WARNING-another wall of text ahead. I think two in one day is my limit
I think it's a bit unfair to categorize MGs as cheaply made or "junk." You have to get in the mindset of the era and mindset that they came out of. They are not a car for someone who wants tons of power, and definitely not a car for someone who doesn't want to get their hands dirty with their own repairs. With that said, the latter point is true of pretty much ANY old car-carburetors, distributors, and the like aren't parts you can just forget about like with a modern computerized car. When people talk about a "tune-up" now, they usually just mean replacing spark plugs. Old cars need periodic tune-ups(they needed them when new, regardless of where they were made) that consisted of jobs like setting the points, the ignition timing, and adjusting the carburetor mixture and idle.
British cars in general, though, have their peculiarities, and some people choose to live with them while others can't stand them.
In a broader sense, though, the MG marque has an important place in the history of what we now call the sportscar.
Basically, someone by the name of Cecil Kimber back in the 1920s started more or less hot-rodding Morris cars to increase performance. With the factory's blessing, "Morris Garages" cars came about, and started building what could be called some of the earliest sportscars. Prewar MGs were often quite advanced, with sophisticated for the time suspension and even overhead cam engines(in the P-type and J-type Midget).
In the late 1930s, WWII was looming, and MG designed a new model on a bit of shoestring budget. Since their engine tooling was worn out, they bought and completely transplanted an engine plant from France, and with that tooling built the reliable but somewhat less sophisticated XPAG OHV engines.
WWII happened with mostly devastating effects on the British economy, but for the British Motor Corporation(BMC) found some salvation in the MG brand and that American servicemen fell in love with the T-type MGs and started taking them back home. The TC pretty quickly turned into the TD, which was primarily made for the American market and sold decently well in what was starting to be a prosperous early 1950s postwar US economy. This was especially significant in that it brought a lot of cash(at least in one industry) into what was a fairly stagnant postwar European economy.
BMC had a big cash infusion from that along with the Morris Minor(which sold well domestically and internationally), and from that developed a new "corporate" line of OHV engines, the A series engine and the B series engine. These were essentially identical engines, with the B series being larger. Initially, displacements were ~1000cc for the A and ~1500cc for the B engine. The A engine made its way into the Minor(to replace its ancient, crude flathead), and into one of BMCs first real groundbreaking designs-the Mini. The Mini was a practical family hauler with a compact transverse front wheel drive layout and handling like a go-kart.
One of the first applications for the B engine was for a totally new and redesigned MG-the MGA. The MGA used sleeker "low" styling(compared to the "upright" design of the T-type) that was similar but distinct from the contemporary Triumph TR3 and Austin Healey 100 series, had a very nice double-A arm front suspension, and was generally an all around excellent product. I still consider the MGA one of the most beautiful cars ever produced. The MGA was also a bit of an experimental bed, with a very nice(but at the time very temperamental) dual overhead cam engine(billed as the "Twin Cam"). Unfortunately, that engine got a very bad reputation that it never was able to shake. I've talked to someone local to me who has a Twin Cam in his garage with a standard OHV engine, and the original TC engine sitting next to it. He'd part with the engine for $10K(which is high, but not out of line) but won't part with the car. I hate to see the two split up-the cars themselves have some otherwise unique features(for the MGA) including 4-wheel disk brakes and really interesting/fascinating Dunlop center lock spline-drive steel disk wheels.
The success of the MGA led to the redesigned MGB in 1962, which improved the suspension, changed to a unibody, upped the displacement to 1800cc and consequently the power to around 90hp, and overall made a very good design even better. The MGB was an overwhelming success, especially in the export market. It sold around 500,000 from 1962-1980, and 450,000 of those came to North America. There were also some derivative designs. The 6 cylinder MGC was a bit underwhelming-it was a lot more powerful, but the big 700lb cast iron inline 6 hanging out over the front axle really upset the handling. Postwar MGs had a reputation for being less powerful than the competing Triumph models, but a fair bit more nimble and better handling. The MGC was supposed to go right up against the Triumph TR6, but it was slower in a straight line and handled worse. Some of the body modifications to fit the 6 cylinder resulted in some weird and sort of unbalanced lines on the hood(including a big bulge on one side to clear one of the carburetors, which almost gives the look of someone realizing it wouldn't close, banging a dimple in it with a hammer, and saying "that'll do"). The real winner was the MGB GT V8, which used the aluminum Buick-Rover V8 that didn't change the weight distribution or overall weight appreciably. Unfortunately, it never was exported to the US(some are trickling over now)
Time, unfortunately, wasn't kind to the MGB either. When BMC and Leyland Motors-two previously rival companies-merged in the late 1960s to become British Leyland(BL), a lot of the BMC marques were left to founder in favor of the Leyland marques. Triumph received somewhat more attention than MG. The MGB was intended to only have a production life of ~10 years, but the money for its intended replacement never materialized and instead was "wedged" into Triumph to design the much-maligned TR7. The MGB soldiered on for nearly another 10 years past its intended design life, but with a bunch of changes not necessarily for the better. US crash and pedestrian safety standards caused a complete redesign of the front and rear bumpers to an all-rubber design("5 mph bumpers", or bumpers that wouldn't be damaged by a 5mph impact against a solid barrier) and a raised ride height. This was done for the 1974 1/2 m/y. All US cars were held to the same standards, but because the US was MG's biggest market and it required a non-trivial redesign of both the undercarriage and sheet metal, all models were modified. Pollution laws crippled US market cars even further-first with a low compression engine in 1972(8.0:1 vs. the previous 8.8:1), followed by California's prohibition on manual chokes in that went into effect in 1975(which caused a change to a single Zenith-Stromberg carburetor, something significantly easier to auto-choke than the dual SUs used previously), and then in 1976 the requirement for a catalytic converter brought in a budget cocktail-napkin designed combined intake and exhaust manifold that was incredibly poor flowing even if the cat weren't in it. Oh, along with that, a somewhat common mode of failure for the carburetor caused the engine to run overly rich(which would make the catalytic converter glow red hot) and the carburetor would leak gasoline right onto the cat, which set more than one on fire. The late ones are not very desirable or valuable today.
The A and B series engines really are, in their own way, great engines. They were made to be used in everything from sportscars to utility vans, and consequently aren't perfect at either task. At the same time, in a lot of ways they are VERY over-engineered with the philosophy of "if some metal is good, more is better." It's worth mentioning that they were very long lived designs as well. The A engine was used in the Mini into the 90s in England, while the B engine soldiered on about the same amount of time in India in the Hindustan Ambassador(which started as the Morris Oxford in the 1950s and was produced until 2014). An early MGB "angle cut" connecting rod is heavier than one out of a Chevy big block 454. I remember taking a pushrod to our machinist at work to ask him if he could shorten it-before he knew I was into British cars he spent a minute or two looking it over and said "I don't know what this is out of, but it has to be British." When I confirmed, he said "I've never seen a pushrod this size that was this heavy or solid." This wasn't unique to MGs, but British postwar customers often preferred simpler and easier to understand controls, which meant that manual chokes stuck around a whole lot longer than on pretty much any American made car. MGs never had power steering, and didn't get any kind of power assist on the brakes until 1975. Power windows were out of the question-for that matter 1962 ads for the MGB touted "Roll up windows" as a feature. That seems strange now, but it was a big deal when you considered that MGA windows(and other British contemporary cars like the Healey 3000) at the time still used manually-attached side curtains rather than a roll up window.
There are perceived issues that plague pretty much all British cars of the era. Electrical systems in particular have a bad reputation, and it's often blamed on Lucas, which was the electrical supplier to virtually the entire British automotive industry. Most cars use simple, straightforward, and minimalist wiring that many over the years have tried to "improve" much to their detriment. The single biggest issue is poor grounds, something that plagues virtually any old cars. To an American mechanic use to working on big Holley and other similar carburetors, the SU carburetors used on a lot of British cars seem utterly incomprehensible and having two of them induces nightmares. In reality, SUs are simple and if properly maintained are reliable. They also can-at least in theory-meter fuel significantly more accurately than any other carburetor design(you'll find statements to this effect in Bosch literature on early fuel injection systems). They just work very differently from a typical American carb. Syncing two carbs properly takes a bit of work, but it's second nature once you're use to it.
The oil leak thing is also a running joke. Truth be told, A and B series engines with good condition seals all around are tight and don't leak an appreciable amount of oil. The old XPAG engines used a rope rear main seal that will reliably leak several ounces of oil every time the engine is shut off, but that's not a unique design. In general, though, with a lot British designs, building things to somewhat looser tolerances allowed them to be a lot more forgiving of poor maintenance, and some of that leads to just accepting oil leaks as normal even though they're not in all designs. A worn out MG engine will use a lot of oil, but then so will any other worn out engine regardless of where it's made(worn rings in particular cause issues).
England in the 1960s and 70s was known for significant labor unrest, and that often crept into poor quality control. It's hard to even build anything when your workers walk out 400-and-some odd times in a year! The Leyland and BMC merger also was less than perfectly executed, with a lot of needless redundancy and also in some cases open hostility between marques in the workforce. To the credit of MG, workers at the Abingdon plant(which is where the vast majority of MGs were built) were mostly content and focused on building good quality products. At the same time, they were at the mercy of supplies from a lot of other plants, and no matter how good the workforce was in that one plant, it was hard to consistently build good quality cars when you had inconsistent quality in other parts coming in.
The British auto industry was known for not necessarily using the best steel or for adequately rustproofing it, so rust is a constant problem on pretty much any British car of that age.
Still, though, pretty much anyone who owns and old Little British Car(LBC) will tell you that they are absolutely in love with them. Yes, there are quirks, but one of the descriptions owners will often use is that the cars have character. Hop in a good running MG(or Triumph or whatever else) on a nice day and find a curvy country road. The lack of power assist and the like makes you feel connected to the road in ways that no modern car can. The fact that they're not overly powerful both keeps you out of trouble, but also keeps you on your toes with watching the road and things like proactive gear changes-and by the way the gearboxes tend to have such tight and precise shifts that shifting really is a pleasure. The "wind in your hair" feel from low windshields is hard to match from anything short of a motorcycle. It's not everyone's "thing", but it's hard for the driving experience in a situation like that to not put a smile on your face.
As for values-the MGB is something of a victim of its own success. Early ones, especially with a proper restoration or good original condition(note that there are a lot of bad restorations out there, but you need some knowledge to recognize what's right and what's wrong on the early cars) are bringing stronger prices. The rubber bumper cars have too many black marks against them, and one needs to be really strong in a lot of areas to get even a mid-4-figure price. I'm partial to the 74 1/2 GT(hatchback). Export of GTs stopped in 1975, and only a bit over 1000 rubber bumper ones made it to the US. I actually nearly bought one a few years back-it was a few miles up the road and could have been mine probably for $3K, but the clutch was slipping badly and I didn't think I could drive it to Cincinatti where I have a friend who's willing and able to provide the space and help with the job. That aside, though, in general Bs are just too plentiful. Nice MGAs(even the common 1500 models), TDs, and TFs still are $15-20K(if not higher), and prewar MGs bring really strong prices when they come on the market.
The bigger 6 cylinder British sports cars both are more desirable just for being more powerful and also less common. There's also the "halo effect" of the car everyone wants, the Jaguar XKE(E-Type outside the US). They're sleek, fast, and much more luxurious than most any other British sportscar. Turn key XKEs are solidly in 6 figure territory now, and even good starting points bring higher 5 figure prices. Since they've become mostly unobtainable, the Big Healeys(particularly the 100-6 and 3000) have crept up and really nice ones are flirting with $100K now(10 years ago, $30K would buy a nice if not concourse level one). TR6s are being pulled up now, and even the negatives of the MGC are kind of being offset by their rarity and I've seen more than a few MGCs change hands at $15-20K.
In all of this, I haven't touched on the other major 60s and 70s MG model-the MG Midget. The Midget name went back to the early days of the marque, but went away in the early postwar years only to reappear in the 1960s in a "shrunken" unibody MG. They're tiny, but are even tighter handling than the MGB. They were powered by the A series engine-later in a really zippy 1275cc displacement. Unfortunately, the rubber bumpers kind of killed them. They picked up the really crude 1500cc Triumph engine, and few folks really even like those cars-they handle worse than the older ones, are heavier, and MG guys don't like the idea of having a Triumph engine up front. Late Midgets are virtually give-away cars, while there is some value in earlier ones.