Linux Gaming

At that price point you might consider getting a Steam Deck. Although getting a PC should give you more power.

Does it have to be a mini PC? For $600 you could assemble a pretty decent full size PC with performance levels similar to or ahead of current gen consoles.

For distro, I can recommend Fedora (KDE Spin if you care about VRR) as it strikes a good balance between reliability up-to-date packages (fairly recent kernels).

EDIT: If you need to save space, consider how much you need to save. You could go for an ITX build for example, which can be quite compact (although bigger than what I would consider a mini PC).

I've discovered Asrock Deskmini X300 which looks like what I want as my NAS. Might be worth checking it out.

Beelink is nice and cheap

If you want the "cheap but can still play 2D games/older emulation (wii yes, switch no, others in between)", you could look for used SFF office PCs, where I live they are less than $300 Canadian ($200US).

If you want decent punch for the uses you've stated, a deskmini with a good APU (AMD Ryzen 3400G for example but there are many newer and improved ones out there) will fit your bill.

Graphics cards eat up budgets quickly so only consider them if you want to spend at least $900 total in order to play very new games at high settings.

With that budget I'd personally go for one of these and just put Mint with the EDGE image on it. Ubuntu 24.04 is coming out in a couple months which means Mint 22 will be out a couple months after that, both using the 6.8 kernel.

I have been looking at the Asus MiniPC series, possibly the NUC series as well that way you can get an external GPU later if needed