Just a note.
Game mode uses a Wayland compositor named gamescope which uses the digital rendering manager directly so it can access hardware KMS planes, etc. It doesn't sit inside another compositor.
Desktop mode is x.org (X11) using KDE's compositor kwin.
My observation is that game mode works pretty good both on the internal screen as well as external screen.
Desktop mode, not so much. Sucks. I find it much more straining on the external screen compared to Windows (and especially my other PC). It was better in version 3.3 of steamos, but got worse in 3.4. The converse is true for game mode and I actually found some games better in desktop mode in 3.3, but probably not now.
I have also tried the Windows drivers. They are so-so, but I don't care for them. Will probably be better later though.
3.5 of the OS will have a new kernel, more mesa updates, etc. Will be interesting to see how it changes over time. I don't think Windows support will be touched much this year. Maybe next year.