It's especially frustrating since you always stumble upon it when you have something else in mind, this your flow is broken to re-confirm something you did a month ago.

I'm not aware of a fix, hopefully more people will write a feedback about it.

It seems like they could just give you a notification reminder once a month that apps have that permission, without forcing you to grant it over and over

Does it mean macOS will periodically ask me again and again that Teams/Zoom/etc need screen recording permissions? As if I didn’t have enough pop-ups and prompts in my life already

I think that's the case. It's doing it for Shottr(screenshots app) and Ice(toolbar management app), the two apps I regularly use and require that permission.