The team behind EmuDeck, a project that started off to provide easy emulation on Steam Deck / SteamOS have announced their own hardware with the Bazzite Linux powered EmuDeck Machines.
I've bought a few Gmktek Intel N97 mini-pcs on Amazon (US) for around $200 usually. So far I've only used them for htpc @ 1080p but N97 GPU has a higher boost than N100 GPU so I imagine should do alright for lots of emulators too.
Their project timeline being really short combined with a really low "flexible goal" of $69,780 doesn't give me a lot of confidence in it releasing or shipping on time if it ever does.
Honestly? Just learning how to set up moonlight streaming (it is trivial) and using basically any dongle is probably going to be sufficient for this use case. If you care about latency to the point that is not an option, you are probably already sourcing an FPGA. Still, I am not going to pretend I haven't been thinking about using some spare parts to build another HTPC.
And emudeck? It is really hard to tell where third party tools start and emudeck ends but it is a good tool. Steam Rom Manager is a shitshow (I NEVER managed to get it to acknowledge that Blood Omen for the PS1 was not Blood Omen 2 for the PS2 or whatever it was...) and I have strong feelings about anything that relies so much on retroarch (transphobic and abusive shitstains who bully developers), but I can't deny its usability on a Steam Deck (even if you can get most of that with just es-de).
But still, this is a pretty easy problem to solve and most of the cost is just sourcing the hardware. Not super familiar with bazzite as a distro but it shouldn't take too many mods to get rid of the login screen and boot these into steam big picture mode.