Airplay games ??
I can't load game apps on my AppleTV. I also can't airplay games I have on my phone and iPad.
What am I missing?
What is your model Apple TV? Identify your Apple TV model - Apple Support
Game apps, including Apple Arcade, are supported for Apple TV (4th generation or later), not earlier models.
AirPlaying games is often hindered by a slight buffering lag, causing sound delay compared to the source device, and fast response is similarly affected if you watch the TV screen while operating on the source device. AirPlay is just not suitable for that.
Some games for iPhone and iPad are only available there, without tvOS counterpart. You will have to express interest for a tvOS version to the developer. How to contact an app developer - Apple Support