My last PC was a 5-year-old Dell laptop with integrated graphics. As such, a few games would crash upon launching, and I tagged them as "buggy" so that I'd know. Now that I've upgraded, I've gotten most of them working, but when I try to remove the tag -- whether by right-clicking them in the library or using the "Edit" function within the game, after I delete the tag, it comes right back.

What am I doing wrong?

I find the Edit window tags section very finicky. I don't mess with that unless I need to create new tags. The following works for me:
1. Right-click the game.
2. Hover over Assign Tags to expand that menu.
3. Left-click the tag to assign or clear.
4. Left-click an empty area in the window to close the menu.
5. Left-click the game to go to its Overview page and double-check the assigned tags.

If that doesn't work for you, there may be a problem with your local database.