Ok, first, let me say that I very much appreciate how Apple has supported me with the issue. Apple was patiently on the phone with me for 2 hours straight.
From the Health app issue, the iOS 16 bug developed further involving my Apple Watch, which was apparently no longer communicating with the phone (since the iOS 16 / WatchOS 9 update). The watch cannot ping the phone. We then took steps to unpair the Apple watch. This proved to be a very tedious and complicated process as we tried different ways of shutting down, resetting, unpairing, pairing the watch. The watch eventually unpaired, and could no longer pair successfully with the phone after (the Watch app was hanging during the setup phase and the only way to address it was delete the Watch app from the phone).
Apple Support ran several diagnostic reports on the phone remotely (with my consent of course), but at the end of the 2-hr call we were still unable to resolve the issue. It was escalated within the Support team.
This was last night.
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This morning, one of the 3rd party apps (Waterminder) had an update. As originally identified above, any app feeding into Health app was not working on my phone. Regardless, I went and tried to open the updated app. And it worked!
Then I thought to check all the other 3rd party health apps, they are all now working!
Then I went on to check the Health app, and I got the splash screen (the one i was supposed to see after the updated, i.e. Welcome to Health). The Fitness app is also working now!
It seems to me the the update of the Waterminder triggered the unlocking of whatever wall the Health app had created.
I have left the watch at home, so I will check back later if it will pair. But for now, my Health app is working.