Apple TV issues with MacBook Pro and iPhone 16 Pro Max

There’s always been a slight delay for AirPlay, e.g. screen mirroring is not responsive enough for action based gameplay. There is screen capture, re-encoding, and background buffering. But if it stutters or stops temporarily, then that could point to an issue on the home network, or the sending device is too busy doing other things simultaneously at maximum effort (on the GPU). It can be a steady stream.

Have you tried restarting the router and any optional other powered network gear?