I sometimes connect my iPad Pro to a TV for gaming. I use the Apple HDMI adapter though since I found AirPlay to be really laggy.

First off, iPadOS doesn’t have true external display support. It’s just mirroring the iPad, so you end up with bars on the left and right side (and top and bottom, but those aren’t as noticeable in my opinion) and the game plays on both the TV and iPad screen. It’s fine for emulators since most old games are 4:3. But for newer games, it sucks that the whole TV isn’t being used. The full screen can be utilized. MAME4iOS let’s you set the resolution of the TV and then only displays the game on the TV while the UI and touch controller are on the iPad screen, so it’s possible. I’ve never seen a game on the App Store do that before. The majority of users aren’t connecting their device to a TV to play, so I don’t see devs bothering to implement it.

Second, full controller support makes for a much better experience. Right now I’m playing Spongebob Battle for Bikini Bottom Rehydrated and Bloodstained. Spongebob has full controller support. It’s great sitting on a couch and not having to touch my iPad. Bloodstained is only partial controller support. I need to use my iPad to navigate the menus. It’s annoying and feels like I’m playing on a DS with a second screen.

So yeah, outputting to a TV from an iPad works, but the experience could be better. I see it as a trade-off since I’m paying much less for a game than I would on a console or PC.