I, too, have an excellent OLED LG TV but prefer to watch via the Apple TV for most things, due to the far superior interface and a remote you don’t have to wear glasses to use.
However, my son pointed out that the black level sometimes dips up and down slightly while watching Apple TV(+) via the Apple TV box, but not via the TV’s Apple TV(+) app. He’s right, but it is probably imperceptible for most people.
However, the main disappointment here in the UK is the BBC iPlayer app for the Apple TV. The BBC usually transmits HD pictures but it has this fantastic catch-up app, which most people use to watch TV now. Some programmes are shown in UHD (TV’s 4K) but this facility is only available on the TV’s app, not the Apple TV’s iPlayer app, which is extremely disappointing.