I'm starting to think it's a phone issue. Seems like the 7 Plus has the most issues. I have had my 7 for 3 weeks now, and not one drop from LTE to 3G.
it is very tempting to blame the hardware of the phone. you said you've been through several phones - you are not that unlucky.
you also said that by changing versions of the software, the problem appeared or disappeared = software problem
people get frustrated and blame the hardware because going into privacy - diagnostics is scary and they don't understand how the software works with the hardware. all understandable
they also don't know how to dfu restore the phone and how to troubleshoot that whole thing; why icloud backup is better than itunes for restoring; that if you continue to have problems like you are having you can sign into icloud to get contacts but it's better to set it up as a new iphone
you're googling for people with the problem, and you'll find it because there are a zillion phones out there and very few people who know what they're talking about
i realize this is probably entirely unhelpful because im saying no, it's not the hardware, it's the software, and that's not what you want to hear - but if i am correct, that means that if you dfu restore (search for the apple way article) and set up your iphone as new, you will have to redownload your apps but you can still sign into icloud which will redownload your contacts - you will have a phone that's good as new in an hour (assuming you can figure out the dfu thing lol - just kidding - you can.)