I posted a little while ago about possibly getting a laptop that's not as basic as what I have, and have been wondering what the situation is with desktops at the price range I'm looking at, seeing as I'm not taking it anywhere anytime soon (and my current PC has cooling issues and is old).

What I'm looking for:
* Budget: ~$500 or so
* CPU should be at least an Intel i3 or the AMD equivalent (don't want a Celeron/Atom/Pentium this time)
* Must not have a Nvidia GPU
* Games to play: Hollow Knight, Ikenfell, The Alliance Alive, maybe Bard's Tale 4 and PCSX2 (emulator).
* Can either be (phyiscally) a full size computer or a mini one (but not one of those $100-$150 Celeron/Atom ones)
* Must support Linux well using upstream kernel
* (Does not need to come with Windows, as I won't be using it, but it needs to not be a Mac)

So, what does the situation look like here?

Would one of these work for you? https://www.anandtech.com/show/15947/asus-pn50-mini-pc-with-ryzen-mobile-4000-apus-coming-september

(RAM & Storage not included in price)

I guess you can find Intel NUCs in a similar price range. Also shuttle mini PCs but the cheapest ones tend to be Celerons. An i3 might be just about reachable at that price.

Personally I would stay clear of anything Intel and anything pre-built as they give you less power for the same amount of money.

Why not build one yourself and save some?

Just as an example taken from Amazon:
AMD Ryzen 5 3400G------------------------------------------------150USD
A B450M mammaboard----------------------------------------------80USD
Corsair Vengeanc (2x8GB) DDR4 DRAM 3000MHz------------60USD
Corsair CV550--------------------------------------------------------70USD
Samsung 860 EVO 500GB-----------------------------------------70USD
Corsair Carbide Series 175R--------------------------------------60USD

= ca 490USD

Alternative pre-built AMD computer for 430USD (cheapest pre-built I could find, but has less):
https://www.amazon.com/Gaming-Computer-Channel-Graphics-Windows/dp/B08FRSBT6B

Both are well within (and more) of what PCSX2 needs and your budget, and you can already see how much more you get by reusing components and/or building yourself, but there's no problem using Linux on either of these.

EDIT: Found something cheaper and still AMD...

I would rather not have to deal with installing the CPU, applying thermal paste, and installing the cooler properly.

(There's also issues with cable management.)

If you're okay with a mini-ITX form factor, the System76 Meerkat might fit your needs ($550 for a basic config):
https://system76.com/desktops/meerkat

One feature you may be particularly interested in is that System76 *exclusively* builds Linux boxes (you can't even select Windows as an option). All of their boxes come with your choice of Ubuntu LTS or their own Ubuntu-based distro (Pop!_OS) -- which means they provide and update a full set of Linux drivers for all their hardware. One of the biggest issues you'll have with pre-builts is trying to ensure Linux drivers exist for all the components. System76 only includes components with active Linux driver support.

Their lowest priced mid-tower desktop is $800, but it's quite nice if you can stretch your budget that much:
https://system76.com/desktops/thelio-r1/configure

If you select the custom build option, I believe that you will find an appropriate Lyte Custom PC which will meet your needs.

I guess it won't sway you but fwiw retail CPUs that come with a cooler tend to have paste pre-applied to the cooler. It pretty much looks and behaves like a sticker, just put the cooler on the cpu, tighten, you're done. See pic. Installing a CPU boils down to open a lever, drop the CPU in the socket (checking the triangle in the corner for correct orientation), close the lever, done. Barely harder than plugging a type-A USB device :)

Depending on what hardware you pick, you might not need much cable management at all. For example with an mSATA or M.2 drive (plugs directly to the motherboard) and integrated GPU, the only things you'll need is power to motherboard, fan(s) to motherboard fan headers, plus front panel power/reset/usb/(optionally audio) wires to their respective headers. That's less cable management that I need behind the PC case after I've plugged in audio, ethernet, power, keyboard, mouse, optical drive, monitor and all that :<

I remember, when I built my current computer (and probably didn't apply the thermal paste correctly, as it got hot enough to throttle under heavy load even when the cooler was still working properly), that installing the CPU required an uncommon type of screwdriver that I didn't have and I had to go get one, which was rather frustrating.

Nothing wrong in going pre built, if you manage to find a nice deal a lot of money can be saved vs custom built.
However, going custom built you can ask the shop to assemble all the parts for you at a resonably cost. On my country such assembled computers come with FreeDos installed.

Any modern "gaming" desktop part can run PCSX2, although not in the most accurate setting. I can play Shadow of the Colossus (one of the hard ones) on my old 2gen Intel laptop (i5-2520m).

To play stated games forget Intel UHD graphics, they are weak and 5 years old.
For integrated graphic solution, you either want Intel Iris pro, wich usually are only found in quite expensive CPU's or AMD Vega 8/10/11. Either of these options will still be weak but enough if you don't mind tweaking the game settings and play at lower resolution, say 720p.
You also want dual channel RAM (two sticks), this will get ~30% performance and anything helps with these kind of integrated graphics.

Otherwise, the option are going dedicated graphics or wait until next Ryzen lineup launches. Fortunately the AMD RX550 finally dropped the price for a more realistic 60€.

Some interesting mini-pc links
udoo bolt
simply nuc
AMD list

I've actually been looking at that udoo bolt gear. Looks reasonably priced (but note that the price does not include RAM or storage), and one thing I like is that it has modern connectors (USB-C with Displayport and USB Power Delivery).

(The included Arduino Leonardo compatible microcontroller is interesting, and I can think of uses for it (like connecting an IR sensor to allow the computer to be woken up with a remote), but it does seem more like a novelty and I probably wouldn't use it much, I'm thinking.)

There is this pre built I found
https://www.dell.com/en-us/shop/desktop-computers/inspiron-desktop/spd/inspiron-3880-desktop/nd3880eejks
Good CPU, intergrated graphics so as long as you aren't running heavy games.
The case I don't think has room for a GPU upgrade.

To play the OP stated games it will need a dedicated GPU. Note that Bard's Tale 4 recomends a GTX 760/Hd 7970 as minimum requirement and that should be at least 5 times the integrated GPU power.

If you don't count Bard's Tale 4 (which may be an outlier in terms of system requirements, compared to the other games I'm interested in), how much does that change things?