What kind of gameplay do you have in mind? I’m guessing a puzzle-type game (like a room escape), but you could honestly do a number of different things (tower defense? Platformer?).
I think the answer to your original question largely depends on this. Did you have anything else in mind about the experience?
Doubtful. Without Linux running the majority of the internet, I doubt the desktop scene would look the same as it does today.
I’m not an expert, but any time I’ve needed to do this, I set up my own router as a client to the parent router, and I set my router (client) as the DMZ in the parent router. Effectively you end up with two routers that are both (more or less) connected directly to the internet, without the two networks messing with each other. It’s also minimally invasive to the parent router (even old stock firmware has always had a DMZ option).
The tricky part then is using the wireless connection as your “WAN port,” rather than a physical one. In which case, as long as you can install OpenWRT on it, you should be fine.
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If you feel most familiar with Debian, I recommend looking at Mint. It has great compatibility and is otherwise easy to use, and it handles GPUs well (including Nvidia). In terms of hardware, I’ll agree with the others here–used is the way to go, you’ll get a lot more for your money than buying new.
That being said, you can also look for clearance and/or refurbished PCs. This one would probably be powerful enough.