Edit: “Updating to the legacy 580xx drivers doesn’t show me a desktop anymore”, just in case someone else can stumble upon this by searching something similar.
Thanks to @deadcade@lemmy.deadca.de and @Ooops@feddit.org I did try fixing it out of curiosity. I had forgotten to install linux-headers. Hopefully someone who actually has the same problem as me, and needs to fix it, can use the tips given in the comments. On my end, I just had to install linux-headers and one reboot later it worked.
Always check if you have all needed packages and don’t just “remember” that you had them installed.
While the distinction can be important, the snapshots from right before the update are exactly what you want in this case over some actual but always somewhat outdated real backup