This is what I've been playing around with recently- the Razer Edge. It used to go for $400, but they slashed the price a few months ago to $250 and it's great for that price. It's a little phone-sized tablet that comes with a Razer Kishi v2 Pro controller, which iirc is $100 by itself. Specs are pretty good and it has little to no trouble running games up to PS2 and Game Cube. Saturn runs well too. The screen is an AMOLED 2400x1080 144hz display. It's gorgeous and super smooth, although for emulation there's really no point at using 144hz so I usually drop it down to 60hz. It has enough power that you can run a lot of games at crazy high resolutions- PSP at 1440P on here looks really slick! Some emulators also allow for widescreen and even super widescreen options, like Flycast.
It can also do video-out over USB-C up to 4K 60hz. I had some issues with this on portable monitors (mainly issues charging and doing video out at the same time), but no issues with an hdmi dock connected to a TV. A lot of these handhelds tend to have limited or shitty video-out features, so this working so well is welcome. Something else that's pretty cool is that when using Android's developer options for desktop mode, you get desktop style versions of apps, I.E. browsers with tabs that work like s full OS browser and not a mobile version.
There were some issues with this thing at launch including micro sd cards being restricted to fat32, meaning files over 4gb wouldn't work, but these have been fixed with system updates and now exfat works. Just something to keep in mind if you're looking at reviews on it from 2023.