As far as cases like the PS4 goes, closest thing I can think of is the Sentry by Dr. Zaber (which you basically need to pre-order on the Indiegogo page to get your hands on, and requires one to really plan around its unique setup to get something that'd work. Unfortunately, none of these cases have ever appeared on PCPartPicker last I checked, which would have been recently) or the Fractal Design Node 202...which is more of the size of an OG Xbox One, except with more of a reason to be the size that it is. The only other option would be the Silverstone RVZ02/03. You could even go with something so small that it had me envious of it like the Ghost S1.
But let me paint a picture for you as to how outmatched your price point is to your expectations, assuming that nothing in this build is coming from a previous build, which you seem to imply, but haven't outright stated, so please correct me if I'm wrong in that assumption:
https://pcpartpicker.com/user/Silent_Gunner/saved/DgbkJx
When I went to go and to a conversion of 500 pounds to USD, it came out to $686.80. Now, PCPartPicker doesn't have the i7-9850H even listed. And here's where I should stop and inform you that, while I have never personally used the and i7 CPU that goes beyond the x7x0 numbers in a given Intel CPU generation, I have read that those CPUs have compromises and don't support some of the standards that, say, a full fledged i7-9700k would, in this case. It's something I read on the PCSX2 forums long ago back when I was building my first PC.
Personally speaking, it sounds like what you want simply doesn't exist. Android TV boxes aren't sold to enthusiast gamers like you and I; they're sold to a family or someone who doesn't have a smart TV and wants to add "smart" features to it. They aren't decked out with some x86 processor; if anything, its some mobile chip or an ARM processor that simply isn't strong enough to approach what a full-fledged PC can, and while it may get the emulation capabilities to do so one day, that day is very far off from even being something that I would classify as being practical as opposed to experimental like Dolphin is on the Nvidia Shield and the Nintendo Switch.
Are you looking for an AIO box to do everything? Or are you just wanting to be able to play everything in HD? Because a PC running an Intel or Ryzen chip like what you got listed is your only bet. With the latter, all you need are the OG systems themselves, and a willingness to figure out how to homebrew them, convert their video signals to HD, and everything else in between, which is easier to do nowadays compared to how it used to be thanks to various devices, like ODEs for the GC, Saturn, DC, PS1, and the ability to run games off of an OG Xbox HDD. To play these games in HD, you have options like the RetroTINK to go and upscale these old games to TVs and to convert their video signals to an HDMI connection. You might have to be willing to invest in some switchers and a bunch of cables if you want to do this, tho. And have a bunch of controllers, memory cards, a bunch of tools to disassemble old consoles that only work with those old systems specifically, etc..
All of this is gonna require some introspection and research on your part. I can't make that decision for you. I would strongly encourage you to grab your favorite drink, play some chill music playlist, open a bunch of tabs on your current PC, and do some research on the more demanding emulators and what the requirements are to run those games. Or, if you decide to just rig a bunch of old systems and have a retro setup of sorts to have a "it just works" solution like what you seem to imply you really want, get a RetroTINK, a switcher, and see what else one should do if they want to get the most out of those systems. In the case of the OGXbox, PS3, Xbox 360, PS2, etc., just get the original systems themselves, (tho avoid the Super Slim PS3s, as I've heard their homebrew options are the most limited out of the three PS3 models. I'd say the regular Slim is the best one tho) as they're easy to modify even nowadays, and we even have forums for those who still use those systems to this day here on GBATemp!
Best of luck, ital!