The default "App Center" does not support flatpaks (only snaps). However, if you install the GNOME software center with the suggested dependencies:
$ sudo apt install --install-suggests gnome-software
Then you will be able to install flatpaks, debs, and snaps.
Oh yes, that's it! I can see App Center and Software. I guess I've installed gnome-software in the past. This was Ubuntu 20.04 originally.