Most smaller handhelds are ARM based with Android as the OS. There are plenty of those in all kinds of form factor and price point available. So if you're fine with emulation you can probably find one to fit your needs. Although x86 emulation on Android also seems to be on the up and about, so you should be able to run at least some less demanding Steam games.

The problem with higher performance (regardless of ARM or x86) is that it needs power and cooling, which need space. That's why the handhelds that can emulate Switch games are much bulkier than the ones that are content with N64 emulation.

If you do want to go through with building something yourself I just found this video of someone ripping a Samsung Flip apart and building his own phone with a keyboard out of the parts. https://www.youtube.com/watch?v=qy_9w_c2ub0 It should give you a good overview over the whole process and the difficulties one might find along the way. He also seems to have videos about building a handheld although that doesn't look as clean as the phone, at least on the thumbnail.

Pair a controller to your cell phone. Done.