I have an original Steam Link in my living room, and I use it to play games from my PC on the couch. Does Steam on Linux even support this?

Answer: Yes.

Is your PC going to be a gaming-centric PC? Or are you planning on doing other things like work or other stuff?

There's plenty of gaming-focused distros out there. Some easier and some harder to understand.

From your hardware You'd probably be good with something like Linux Mint or Zorin OS. They're Ubuntu based and are pretty solid and easy to use. They support your hardware out of the box. Kubuntu might be a good choice too if you want a more windows-y experience with more customization options. I personally have Kubuntu and my 8bitdo controllers work without a hitch via bluetooth. I play mostly indie games and a few AAA titles from Steam, GOG and Epic via the Heroic launcher as well and have had zero issues.

If you REALLY want a gaming-focused distro, you'll have to go with a Fedora based distro then. Something like Nobara.

But you want to avoid less stable distros like Arch-based ones. Like Cachy OS. Those might end up being more of a headache than anything with problems occurring after updates and a malware-infested 3rd party software repo.