Is there a way to search the Google Play store for games compatible with Android TV? I've tried the following link but it is clear to me that there are titles that are compatible that are not on this list. https://play.google.com/store/apps/collection/promotion_3000e24_androidtv_games_all
Go to android play store website and login using the same account you use on tv, then in search field tyle "game" for example. You will see there is now a filter "all devices" - in there You will find your tv and just select it.
Remember that You can use apk extractors on phone and move apps to TV - though I have never actually used this option, so dont know if this actually works
You can try doing a search for games then selecting your android tv under devices.
I've built a small game you could try: https://play.google.com/store/apps/details?id=nl.rwslinkman.trictrac
Edit: It's only available for AndroidTV
That list seems incomplete and inaccurate. For example, "Hotline Miami 1 & 2" are available on Shield TV but missing from that list. "Modern Combat 4: Zero Hour" is on the list but definitely not compatible with Android TV. I sideloaded it once. It doesn't support controller/keyboard and requires multi-touch. So can't really play it with USB mouse either.
I don't think there's an easy way to find out a complete list. Even for a compatible game, it doesn't necessarily support all Android TV devices. Some require OpenGL ES 3 and won't work with OpenGL ES 2.x boxes/TV sets. Some have exclusive deal with certain devices, like "The Final Station". I think the best way to find out compatibility is to visit a game's Play store page and check if it's compatible with your devices by clicking on "This app is compatible with all/some/none of your devices", or browsing through Play store on an Android TV device.