The fact that you cannot easily listen on headphones through you phone is a dealbreaker. With Roku you can connect your headphones to you phone, and then just start streaming the audio directly to your headphones. With Apple TV it sounds like you need to use the remote to connect headphones directly to the Apple TV. That's such a pain. Why can't they just make an app.
Connecting AirPods to an Apple TV is literally one-button tap
If you use Apple headphones it works directly without the remote
I use my AirPods with my Apple TV on the daily.
It works with any bluetooth headphones, directly.
Any Airpods connect directly to the AppleTV. You don't need the remote, phone or an app at all.
If you're using wired headphones then a $20 Bluetooth dongle will do the same.