Most of the AppleTV apps are web apps
And like all of their platforms, you have to use the Safari/webkit core for your rendering engine.
The next part is that hostile content needs to be served from somewhere the TV is loading. Ads are an obvious avenue, as would watering hole attacks on the servers hosting popular content, I'd go after the second tier apps for things like news channels.
What I would be more worried about is one of the developer libraries the apps link to going evil. There are too many shady "telemetry" frameworks in these things, and most of these devs don't seem to be doing due diligence their tools. So if one of these tools goes bad, hundreds or thousands of Apps could start telling their clients to request content from anywhere, triggering an attack based on a fault like this.
So even on devices people don't use to browse the browser can still be attacked.