I think you're correct. I have the Gen1 4K TV as well and no FaceTime app. Also - https://www.macrumors.com/2023/06/05/facetime-launching-on-apple-tv-with-iphone-and-ipad-camera-integration/#:~:text=The%20feature%20requires%20a%20second,for%20camera%20and%20microphone%20functionality. Upgrading to Beta 2 did not cause it to appear, so I think that's that.

I added it as well via feedback for my iPhone 14 Pro, and the status says that there are no other recent similar reports. One just need Google it and it's all over.