Ubuntu 21.10 gnome shell login not available

This turned out to be just my confusion with respect to shell extensions, so I vote for removing the whole discussion.

Ubuntu desktop is actually a customized version of Gnome Shell, so you are using it logging in to Gnome Shell.

Moving to a pure Gnome Shell session is possible on Ubuntu 21.10 by installing gnome-session:

sudo apt install gnome-session

Now, you will find options to log into pure Gnome Shell, either on Wayland or Xorg, in the menu you obtained with the cog wheel.

You can also change the colors on the login screen from Ubuntu styled to default Gnome Shell.

On Ubuntu 21.10, run

sudo update-alternatives --config gdm-theme.gresource 

On Ubuntu 20.04, the command is different:

sudo update-alternatives --config gdm3-theme.gresource

Select a different theme from Yaru there.