Why will touch i.d. work only a few times on my iPhone before failing?

I have a 2-year-old iPhone 7. When I got it, touch i.d. worked for about 5 months. Then it started working only intermittently, then only sometimes, and finally it wouldn't work at all. I called Apple Care and was told to upgrade from iOS 10 to 11, and that would "probably" fix the problem. I upgraded, successfully entered a new set of fingerprints and deleted the old set. After that, touch i.d. worked for a few weeks before conking out again. I've since upgraded to iOS 12 and now have the most recent version of iOS 13, but none of the upgrades have fixed the problem: I create a whole new set of prints, touch i.d. works for three or four days, then fails completely. I've tried using different parts of my finger/thumb at various angles, but nothing works.

I guess it could be the hardware, but Apple doesn't seem to have any suggestions, and I've read that many other people have had this happen. It's just a nuisance when you can enter your passcode, but when you're trying to use a ride-share app or pay for something on a site that will only accept touch i.d., it's a big fat problem, not to mention a big fat time suck. Has anyone out there found a permanent solution?