For anyone still experiencing this, I appear to have resolved it (for now at least, I haven't rebooted yet). Open the Snipping Tool from the Windows screen icon (i.e. not Windows key-Shift-S). Then click on the 3 dots at top right (when you mouse over, it says See More). Click on Settings. In my list the second option Automatically save screenshots was set to ON. Change the setting to OFF. Now when I use the Windows key-Shift-S, the screen grab remains on screen and allows me to edit it.
Incredibly, the opposite has fixed it for me. I had turned off autosave, and now that I turn it back on, I get notifications again.
Background: I've tried the default apps fix, the reset/restore, the reinstall, etc. All other things in this thread. Nothing helped.
Turning on autosave fixed it.
I tried turning off autosave again, and I again stopped getting notifications. Turning autosave on and off reliably fixes and breaks the notification functionality.
I guess I'll have to manually clear out these useless files every once and a while, but at least my snipping tool is functioning again.
I hope this post helps others with this incredibly annoying bug.