SteamOS is not available as a general install media for your PC. SteamOS from Valve is based on Archlinux. There are many different base Linux systems other operating systems / distros use them and build a new system around it (just like what Valve did with Archlinux into SteamOS). In general, SteamOS itself does not integrate GOG or any other launcher out of the box. You would need manual installation in desktop mode (normal PC mode) and add them into the Steam as non Steam game. Its just like on your regular PC. But as said, that does not matter, because SteamOS itself is not available for general PC.

But there is Linux distribution that aims to fill that gap, called Bazzite: bazzite.gg Its base system is different from SteamOS, but it is gaming first oriented. It’s quite popular. And integrates other stores as well:

Own games outside of Steam? Lutris (Pre-installed) and other launchers can not only run games from most game stores, but cleanly integrate them into Steam Gaming Mode, including: Xbox Game Pass (via battle.net), EA App, Epic Games Store, GOG.com, itch.io, Rockstar Games Launcher, Ubisoft Connect, your dad’s old CDs, and more!

There was also one called HoloISO, which was much much closer to SteamOS. But that one did not see any development since over a year, so I would rule that out.