SteamDeck is only good for light gaming on the go, you sacrifice a lot to fit into such form-factor, so going any higher that native 720p you have to settle for the low image quality for any half-decent game, which, depending on your taste might kill all enjoyment because it's going to look like crap. That said, nothing beats it when it comes to mobility. There are a number of mini-pc using higher-end laptop SoCs (like AMD 5900HX or 6900HX), but it's price can go quite high depending on how much RAM and SSD storage you want, approaching some "gaming" laptops.
The problem with all those solutions for games is that GPU becomes obsolete much more quickly than other components, and you can't replace it while leaving everything else. With that you have to go for something which can fit a discrete GPU, but those devices - while relatively small compared to a "full" PC - are not something I'd personally lug around with me, so it would only work if you only occasionally need to move it, and it would be mostly stationary. Also those mini-ITX cases tends to be very finicky when it comes to putting together a computer, so I would only recommend self-assembly if you have some experience with PC building, because not all parts will fit all mITX cases (so just because a part fit into a particular mITX case, doesn't neccessary mean it will fit into another mITX case), and getting all cables where they should go sometimes can be a challenge too. At the very least I recommend buying all parts in place(s) which would allow you returning if something doesn't fit. Also mITX parts tend to be more expensive for some reason than their "full" counterparts despite them typically being smaller than the latter.
I would personally perfer going a mini-ITX route because it gives you maximum flexibility when it comes to parts choice while still taking less space than a conventional PC would, but there are some serious gotchas if you don't think it through - like if you put a high-end CPU and a fast GPU, this thing will sound like a jet engine on a take off when loaded up, and yet still probably will thermally throttle - because physics. What is youre budget btw?