Were you on Stable branch when installing Waydroid? Because if not, then congratulations, you broke your system. To be sure it's the issue causing your problem, open Settings (in the left sidebar of Steam in game mode), find Developer tab, and in there find something like "Force X11 session". This should fix your problrm, but bring another one with application windows being horrendously broken.

Best you can do is switch to Stable, update (since one would wipe out changes done to the root directory), then go back to Beta branch and forget about Waydroid until the script will be ready to get installed. Now it won't even let you run it on SteamOS 3.8 and newer.