I think a budget android handheld could work well for game streaming. TechDweeb and Retro Game Corps newest videos show android handhelds that cost less than 100€.

I think even 1cm would be enough. Take a look at the ifixit steamdeck heatsink replacement guide to see how small the heatsink is.

https://www.ifixit.com/Guide/Steam+Deck+Heatsink+Replacement/148904

If you go to chatcontrol.eu you will find more info about chat control and how to fight against it.

Retro Game Corps video My Favorite Handhelds of 2024 will give you an overview of what kinds of handhelds are availible. Altough the ones in the formfactor you want probably don’t have very good compatibility with newer indie PC games.
Winlator is an option to get some PC games to run on android handhelds but it requirers some tinkering. I should add that I am not speaking from experience, I have only watched a few videos.