I figured out what the problem was: Start 10 was dropped from the Win Startup folder. Note that this occurred before I re-installed Start 10 and I never go near the startup folder, so I'm not sure what caused that to happen. It wouldn't surprise me if that's a Microsoft thing.
Reinstall did not add Start 10 to the startup folder, but to answer your question, I didn't "purge" anything, although I didn't see any option other than uninstall.
What is confusing is that I have Start 10 pinned to my taskbar and, if I launch it from there, currently it doesn't replace the Win Start button (it used to).
So, I manually added Start 1o to the startup folder and now it works.
FWIW, Microsoft really doesn't like products like yours and I believe they made things progressively more and more difficult for Classic Shell, to the point where the s/w developer eventually abandoned that product. I suspect the same thing may be happening with Start 10. If so, you should disclose that - if people are buying a product, even for $5, and there is a known, albeit anti-competitive, compatibility issue, that should be disclosed.