Believe it or not, I actually managed to solve the issue in a rather simple way.
What I did was performing a fresh W11 install (in-place upgrade) keeping all files and settings, but rather than getting the latest build via UUP dump I just downloaded the most recent ISO from
Download Windows 11 and just running it.
The fun thing is that now the problematic KB5016629 update doesn't show up anymore, but it's replaced by KB5016691 (which I'm pretty sure I didn't have before).
Maybe it's worth a try if you have similar issues?