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).
And a full size PC will be much cheaper to upgrade later.
An ITX build would increase the costs a bit for motherboard, case, and PSU (an extra $100 or so), but I think the space savings is worth it. A GPU is also an option, and you'd need an external one for a mini PC, which isn't ideal. Get an APU at first, and then a decent GPU later if it doesn't hold up for $200 or so.
This is good advice
Look at logicalincrements.com to budget different levels of hardware
I've discovered Asrock Deskmini X300 which looks like what I want as my NAS. Might be worth checking it out.
I have an A300 I was using for gaming (now it's a home server) and it's great for both. Very small but can fit a NH-L9a-AM4 or (barely) Big Shuriken 3. It has 2x m.2 and 2x Sata but I wish there were more mini-STX cases out there because it can only fit a 2.5mm drive.
Though for NAS DeskMeet might be better.
But aren't there dedicated solutions in this form factor around, with more than 2x 2.5" too?
I would like to know if there are. X300 has 2x m.2 and 2x 2.5" SATA, which is enough for me now, but expandability is low or non-existent...
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.
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.
This there's a whole community of sff optiplex builders out there
Beelink is nice and cheap
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
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