Just an update. Again, I have an iPhone 7 with 10.2 .1 (14D27), 256GB with over 100GB available). AT&T service.
In addition to just about every other thing mentioned here and by both Apple and AT&T support people,
1) I had tried deleting all messages (and clearing all) on several occasions. It did not resolve my issue.
2) I was doing the work-around of trying to beat the inevitable collapse and instead hit the Share button and email myself my message(s) to listen in an alternate medium (can play directly from email on same phone).
I realized last week that I have been diligently doing backups since, and on each new version of the phone, had simply "restored from backup" since 2007, when I got my first iPhone (such a fan boy). Last week, I figured surely a fresh phone with no restored backup must be able to pass the clearly minimal testing the Apple peeps seemed to have done before realizing this latest POS.
Now this was a HUGE pain, and I am only now day-by-day adding back apps and going through the rigor of establishing identities/logging in to each (using encrypted iTunes backup model saves one from all that pain), but at least my voicemail appears to be working. Of course, the "cost" of me having to manually rebuild everything else on my iPhone makes this latest release of the phone app (and its crappy transcription feature) a total bust.