I installed arch Linux in a virtualbox VM to see if that is a general arch Linux problem. But I never got the error message that the launcher can't communicate with steam. I tested it with Gnome on Wayland, Gnome-xorg, both with Chromium not installed, installed but not running and running. Which Desktop/Windowmanager do you use? And are you sure that you used the pacman version of steam and not the flatpak version?
To find out if it is something with your system or something with your user account, it would be good if you try it as a new user without copying over any files from your existing user account.
To find out if it is something with your system or something with your user account, it would be good if you try it as a new user without copying over any files from your existing user account.