I often stream from my steam running on my linux desktop to the steam box connected to my TV and that also work fine.
GPU support is really independent of steam, but these days both Nvidia and AMD have pretty good drivers.
Manage your expectations, I would treat it like a console, you won't be able to play 100% of "PC" games but you will have a very large selection of games including AAA. And in comparison by raw numbers Linux have many more games than regular consoles, not to mention all the emulators you can use.
Performance wise Source engine games run better than on Windows for me and a few Proton (Wine) games runs better too, but that's not the rule as there is a small penalty from the "translation" layers. Vulkan games run very well.
If you try to run all games even the ones that Valve didn't approved to run with Proton ("1-click in the UI and it just works") you will have to tinker and do nerd stuff like using different versions or compiling from source. Don't blame the platform if you go that route without proper knowledge and patience. Also in the same line of thinking don't try to use Wayland yet, specially if you have a NVIDIA card because there's no support from them, and I hope they do something about that but I wouldn't hold my breath.
I thought I had one game with a crash to desktop bug in linux but then I played it on windows and it turns out the crash has been faithfully reproduced.
I rarely even look to see if a game has Linux compatibility at this point, with the exception of multiplayer focused games.
Basically everything I've tried has worked with Proton, with maybe one or two exceptions.