Desktop mode is KDE on X. In that case your comment is true, steam will just run on KDE.
Game mode is a wayland session with gamescope as a WM, but it isn't controlled directly by the user. The steam client controls the WM and exposes those controls to you through menus and handles stuff like notifications and being a driver for the controller, so you could call it a DE in game mode.
More like X session running under XWayland. gamescope doesn't really expose Wayland interfaces to clients (well, technically it does expose some stuff, but I wouldn't qualify that as anything else than implementation detail - although that may have started to change recently, I've seen some stuff being done around xdg-shell there).
Can't find any "kde" or "plasma" references in the process list under normal operation. It's only after you reboot into the desktop mode that you start seeing process names like "plasmashell", "kded5", "startplasma-x11" and so on.
I'm not entirely convinced that using SteamOS can really be counted as "using KDE" in any way.
Well, I wouldn't make that statement so strong. KDE Plasma is the default desktop environment on SteamOS, it's just that it doesn't boot into its desktop environment by default (but you're two buttons away from switching into it).
Steam runs on KDE on SteamOS