This is kinda old, but when I first got my Shield TV I put together a little list of games that work well. These games are all from the Amazon app store because they were free there and not free on the Play store.

https://forums.geforce.com/default/topic/903895/amazon-underground/

I didn't try sideloading it but it works perfectly with game pads so it should work fine: Crashlands. It's similar to Don't Starve and that kind of survival games but with a storyline and missions you can accomplish. It's not free but it's really good and it has cloud saves so you could play in your phone or tablet and continue in your TV.