Jumping back in after a long delay due to getting too busy with work to dig into diagnostics (I was rarely even in front of my personal Mac having the issues), I can report that it appears to be fixed now. At least, I haven't had a reoccurrence in a few weeks, which is past the point of coincidence.
Not exactly sure what fixed it, unfortunately; the most likely possibility is 11.2 (or an 11.2.x update), but it could also have been an iOS update to nearby phones, or the combination of both.
Interesting, and it seems unlikely coincidental, as of 11.2.something my MBP is now always showing my iMac, both iPhones in the house, and my Apple TV in the Bluetooth pref pane, regardless of connection status. I assume this is because I have Handoff enabled (although I refused the connection when prompted for my iMac).
Since this started working properly (and named devices started appearing) without my ever touching the "Allow Handoff" setting, it seems believable that malfunctioning Handoff was the cause and Apple fixed whatever wasn't working with a software update.
It would also explain why my work computer never had problems; it has Allow Handoff enabled, but isn't logged into my AppleID so wouldn't be attempting to connect to anything.
And at this point Apple is telling me to bring the MB to an Apple authorized service, which is almost an hour drive. Also, I have to drop the MB, and pick it up a few days later. Almost four hours wasted, just with transporting the MB. Come on Apple! I get better customer service from Aliexpress.
I just wanted to comment that it's both weird and a shame that you had this kind of experience with Apple Support, since it's so different from my own.
I live far from an Apple store, and Apple has ALWAYS offered me the option of 2-day-FedEx'ing an empty box to put a laptop in, which is then shipped to Texas, repaired, and 2-day'd back to me. I had my current MBP serviced twice for completely unrelated issues, and both times it was back in my hands in less than a week, and I only left my house to drop it off at a FedEx office (and I could have had it picked up at my door if I'd wanted).
A co-worker sent their quite old (
long out of warranty) MBP in for the screen coating issue, and it was the same thing (Apple even apparently replaced the BT module as well since they noticed it was acting up).