This evening I used the Home app on my Apple TV to turn on a light. When I tried to turn it off with the Home app on an iPhone it reported “no response”. Same with a Siri request to a Homepod.
When I went back to the Home app on the Apple TV it allowed me to turn the light off. I could then control it again with an iPhone.
Strange!
Lots of problems with Home and related apps: Owning an Apple Home: The broken promise of Matter
I’m not aware of an Apple Home app on tvOS, did you mean Siri or a 3rd party app? Most problems I’ve had with my non-bridged HomeKit setup have been faulty devices (the original Leviton non-dimming wall switch primarily) or bluetooth devices too far from the hub/Apple TV. Upgrading to the current Apple TV with Thread (Dragonriders of Pern anyone?) to run a motion detector solved my range problem.
The Home app on the TV is available in the upper right hand corner of the screen, near the settings and other controls. (It only show up if you invoke it with a long press of the 
button on the remote and then select the home icon and then the appropriate scene.
I was unable to repeat Micheal’s problem. On my Apple TV, I see the various scenes, not rooms nor specific devices, but I can control them from the TV and the corresponding devices go on or off. It synchronizes with my iPhone without any issues.
It’s the same for me. I also see scenes (and my cameras), not individual outlets/lights/sensors, but after turning on a scene on the TV I was immediately able to turn off the device on my phone.
I posted a bit too quickly without doing more experiments. Firstly, the ATV Home app only displays Scenes, as others have noted.
I have a scene to turn a reading light on. The other day I used this scene and the light came on. When I tried to use Home on the iPhone to turn it off it was reported as unresponsive. But I could still use the ATV to turn it on and off.
This morning I went through the same process and the light was no longer unresponsive on the iPhone.
So a temporary glitch!
Yay - temporary glitches are the best kind :-D. This reminds me of a Homekit limitation I’ve heard about with trying to use scenes like switches, but I forget the details so I’m not going to guess. And thanks to JB for the pointer, including icon! I tried it–nice feature.
Off Topic - From one of Microsoft’s ‘Tips Of The Day’ that used to pop up when Word, Excel or PowerPoint was first launched, giving tips on how to use the software more efficiently (e.g., suggestions on formatting, how to use menus, etc.) and I quote verbatim from Microsoft: “Things that go away by themselves can come back by themselves.”
It really instills confidence in Microsoft applications, doesn’t it? 