See if this is caused by hardware acceleration.

In Thunderbird Menu app icon > Preferences/ Options (depending upon version) > General Scroll to bottom uncheck 'Use hardware acceleration when available'

Exit Thunderbird and wait a few moments for background processes to complete, then start Thunderbird.

Also check your Graphics drivers - are they up to date.

Like I said I can't open settings in this state, but I remember changing hardware acceleration some other way before and it didn't help.

Graphics drivers are fairly up to date.

The brand new Thunderbird 102.0 is affected too. Display drivers have been updated meanwhile.

I don't think anything changed in the brokenness from previous versions; the right side hamburger menu doesn't work at all, the left sidebar is empty and without function, the top menubar mostly works, account settings show all accounts correctly. See screenshot in my previous reply.

Any new ideas?

Huh, after downgrading back to 68.12.1 I noticed the built-in updater (via the About window) was working again, which it previously didn't (it showed an empty update channel instead of "release"). Updates installed this way seem to work fine, meaning I can get working versions 78.14.0 and 91.11.0 by doing it twice. But then running the installer for 102.0 for a manual update breaks things again.

I surmise that running installer packages produces a broken version while the updater produces working versions. What could be the cause of that?

This is a wild guess.

I have heard on rare occasions of a basically crippled user interface like you describe occurring because the installed version was using a language pack that had not been updated for some reason. Perhaps try and English version and see if it installs correctly. if it does remove all of the dictionaries you may have installed and then try your chosen language again. That might narrow where to look here.