And the apps that provide solutions for it, like Bartender, need screen reading permissions which I just can't bring myself to grant.

If you really want to use such a thing, switch to Ice. It’s an open source thing similar to Bartender, before BT was bought by a shifty outfit. It still requires those permissions, but at least you can look at the code. I have a paid Bartender license. I liked it enough to pay for it, but don’t like the road it went down and stopped using it.

Tahoe lets you selectively remove app icons from the menu bar. I’m going to try that for a while and see if I can tolerate not using Ice anymore.