Well, I didn't have the original Cura project saved, so I thought hey, I'll just remake it, since I couldn't avoid that problem before. So, naturally, I can't get the same bug to trigger now. Cura 5.60 slices everything fine, no random unnecessary moves. And since there's no random dancing during infill, both print time and file size is cut down in half.
No clue what the problem was, but it was a fresh update, so maybe so interference between versions that disappeared after restart (doesn't sound very plausible, but stupider things have happened). However, I've noticed that I still have 5.40 version. I've tried to slice the model on it, and lo and behold - the jiggly is back.
So yeah, I guess my problem is solved by using 5.60 instead of 5.40 (even though it was 5.60 version doing the dance yesterday, but whatever). I'm still uploading .3mf file because it would be good to know what exactly could have triggered this issue (some messed up setting somewhere, weird bug caused by orientation, etc.)
Grip project.3mf
IndexGrip.stl