Can the Apple TV remote app turn the TV on and off?

When set up properly, yes. On the Apple TV open settings and go to "remotes and devices" then go down to "home theater control" and there should be a "Control TVs and receivers" option. Select that and follow the instructions.

At some point last year I am pretty sure I was able to use my phone's remote to turn everything on and off since I had the TV set to turn off with the Apple TV. Something in the tvOS updates seems to have messed with that. If you have it set up right and it is not working then it may not be possible. I dont know of any troubleshooting that can be done for that. (You could erase both devices and set them up as new but that seems drastic for this situation)