Unfortunately, recently more and more sites fail to work with Tor Browser. Notably, I am informed by people who are not me, Anna's archive and libgen.

The sites block Tor, not fail to work with Tor browser itself. I know this is a meaningless distinction for end user.

In theory you could use Tor browser with Tor stripped (I heard this is what mullvad browser is?) or go tor-then-proxy (this is what I often do, because I sometimes use whonix at work). I don't know about libgen or Anna's archive, I don't use them.

Why shouldn’t you care and how is the issue with fingerprinting overblown?