You still need X11 or Wayland somehow. I would suggest something that can run in “kiosk” mode. That’s essentially what you want - boot a single app and run that.

If you don’t care about wayland, just install xorg-xinit (since you said Arch) and put this in ~/.xinitrc: exec steam -bigpicture

Run startx to launch it on its own. There may be other dependencies you need for steam or for games, but you don’t need a whole DE.

If you want, you can even enable auto-login and then set up auto-start by adding this to your ~/.bashrc (or whatever shell you use):

if [[ -z $DISPLAY ]]; then
  startx
fi