It's a little hard to decipher what you mean. I think I understood that you basically want to have a headless gaming rig so you can stream your games in the rest of your network. The closest you would get is probably Bazzite, and configure it to directly boot into steam big picture mode so you can use steam link on your connecting devices. If you would like a platform agnostic setup you could look into sunshine. That will involve more tinkering tho.
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Maybe Bazzite because it's geared towards working well with gaming out of the box.
Even "heavy" linux distros are easy to run for a pc capable of running and streaming modern games. Lightweight distros are mostly relevant for older or underpowered hardware.
OK so a lot of people are recommending bazzite. I have been using it for a while and I do not recommend it. Mainly due to the fact you more or less can't install any program other than through the software centre unless you use a vm with the included "box buddy" program which for me has been very hit and miss for which programs work.
Go ahead and try bazzite for yourself but I would recommend either fedora or Linux mint both with the xfce desktop since a lightweight desktop seams to be what your looking for.
distrobox is not a vm
I'd recommend CachyOS if you're trying to squeeze the best gaming performance out of the hardware
should be fairly simple with any distro. personally, id recommend mint or something for ease of use. install steam, and you can use steam link for android (chromecast, roku, firestick, mobile), or steam link for linux (raspberry pi etc). steam has built in compatibility for windows games running on linux, and a built in streaming setup. a bit of an all-in-one solution.
you can try a steamdeck-like distro as others have suggested, but they may be less easy to learn on if youre not too familiar with linux already.
keep in mind nvidia drivers are a sore spot in linux, so if possible try to use an amd graphics card.
Linux uses less resources than windows by default, any is better performance and lighter, cachyos might be easier to tinker with, I got sunshine working on it fairly easy.
You mean as a file server? I'd probably use a minimal Debian installation. But almost any distro should have a minimal flavour.