Is there a SteamOS-style Linux distribution with a non-Steam gaming interface?

You may not need to replace the whole operating system. Some gaming-focused distributions can boot directly into the normal desktop instead of their gaming interface, and Lutris can be installed and configured to launch at login. That gives you a handheld-friendly setup without making Steam the center of the UI.

Avoiding systemd narrows the options considerably. Artix or another non-systemd distribution can be customized with Lutris, Heroic, and an autostart launcher, but you’ll be building and maintaining most of the handheld integration yourself. Expect more troubleshooting than with the usual gaming-focused choices.

The core feature you’re describing is really a controller-friendly game shell, not necessarily a special distribution. A regular Arch-based installation with KDE Bigscreen, Lutris, and shortcuts for Steam and GOG could provide that experience, although it would require manual work for suspend, controller navigation, updates, and automatic launching.

There doesn’t seem to be a mature distribution that matches all of those requirements. The practical route is to install a lightweight Linux distribution, add Lutris or Heroic for game management, and configure a controller-friendly desktop or launcher to start automatically. KDE Plasma Bigscreen or a customized Wayland compositor could work as the front end.