I fixed my game. Here's the thing. The Wicked mod apparently assigned traits or settings to some of my characters, and the outdated version of the mod broke from the update. BUT with the mod removed, the game tried to access these traits/settings and crashed. This also crashed in a new game with a new character, apparently simply because I added this family to the gallery/second save (I tried different ways).
Anyway, updating the game to the latest version + updating Wicked mod to the same version helped me.
What confused me most about this whole situation was that even without the mod, the game still broke essentially because of it. This was something that was hard for me to figure out, so after removing the mods I immediately decided that it wasn't about them.
I don't quite understand how it worked with my tests. Maybe the test with a completely clean The Sims 4 folder worked after all, and I didn't take that into account, added the save file to the folder and thought that everything was fine if I didn't go into that save.