It’s mostly the TV. The input difference between wired and BT should be very small, though the switch is not optimized for wired controllers. The variability of TV response times on the other hand it massive in comparison. Specially modern TVs with heavy post processing who think they are clever trying to interpolate frames or other shit like bad HDR implementations, etc. HDMI DRM also adds latency.

All that causes most TVs to be subpar for gaming. I still game on TV, mostly cozy games. But I accept that nothing competitive will come out of gaming on a TV.