Void and Nix aren't Arch-based.

Which one do we install for gaming

If you have to ask, I recommend Linux Mint. It's not Arch based, which is a good thing because it's going to be really stable and easy for people new to Linux.

Steam is the same regardless of distro because it ships all of its own dependencies, even for Linux games. So if a game works on Arch or SteamOS, it should work on Mint, Fedora, etc.

If you want something that feels like SteamOS, I've heard good things about Bazzite, but my recommendation is still to use Linux Mint and install Steam and Heroic, and then you'll be good to go. I personally use openSUSE Tumbleweed, but again, I recommend Linux Mint for someone new to Linux, because gaming should be nearly identical between distros and Linux Mint has a large community of people to help when you run into issues.