As far as an eGPU, I definitely wouldn't recommend it. It's a nice way to add some umph to a laptop, but generally is going to be pretty inferior to a laptop with a proper GPU in it. It'll run solidly faster if you connect it to an external monitor, versus piping back into the built in monitor on your laptop, but it's never going to be amazing and it's ultimately going to be just as expensive as simply buying a laptop with a 3050 or something mid-low end.
A Steam Deck will work, as it'll run Red Dead 2 at 30FPS without drops; however, once you start getting into the overhead of multiplayer stuff, I can't vouch for that. I've never played RDO, but I do know there's a good amount of overhead running multiplayer games, so it's possible that the game may run perfectly great singleplayer and awful in multiplayer. That said, the folks on Reddit seem to say that it
works well.
Again, I can't vouch for installing custom server tools, but a quick Google suggests it's possible and I wouldn't be surprised if it works just fine. I've installed lots of mods and Windows-first stuff on my Steam Deck. That said, it takes some tinkering and working in Linux, so OP has to be comfortable doing that stuff; adding non-steam games and figuring out working directories that are different from standard Windows. This will be exactly the same for the Steam Machine.
Being me, I'd look for a used Gaming Laptop on eBay, though I have lots of experience with tech and I'm not afraid to crack things open and tinker, so used technology doesn't scare me. So if I weren't me and weren't willing to go second-hand, I'd keep an eye on Microcenter and Newegg to see what I could pick-up. Obviously, things aren't going to be cheap right now, but they're only going to get more expensive, so now is the time to buy.
Here's a few suggestions. I won't vouch for how good they are, but they are "cheap" for what they are and will get your foot in the door.
Both Lenovo and MSI are solid brands and should last awhile.
You could also spend more money, but also get more computer for that money and be able to upgrade it in the future.
There's not really a cheap option here for what you're looking to do. Even second hand, you're likely looking at a minimum of $400 for something a couple of years old and again that comes with all the caveats of used, old tech. That said, if you buy off eBay, you're pretty much covered as long as you buy something that isn't "Parts Only" and is listed as fully working. eBay will cover you and force the seller to refund you if it's not exactly as described.
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