System settings won't open on Ubuntu 20.04 LTS after adding System76/pop PPA

Finally I found a solution!
The cause of this problem was pop-theme, that I installed once, plus its repository entry, that provided me the broken System settings (gnome-control-center). In order to solve this:

1. I installed Ubuntu desktop PPA.

   sudo add-apt-repository ppa:ubuntu-desktop/ppa
   

2. I removed all System76/pop (and other trush) entries, with update-manager, leaving Ubuntu desktop the only one active.

3. Now that I have the proper repository entry added, I re-install gnome-control-center.

   sudo apt install --reinstall gnome-control-center
   

4. Enjoy Gnome System settings!

sudo apt install ppa-purge

For the next step the PPA has to be active.

sudo ppa-purge ppa:system76/pop

This removes packages from this PPA if they are not available from Ubuntu repositories, or reverts to Ubuntu package versions.

I had the same issue. I was able to fix by removing the GNOME extension/theme which caused the issue and then running:

sudo apt install --reinstall gnome-control-center