Regardless of your output device or means, TVs were never designed with low input lag in mind.

As you can see from [this study][1] TV latency varies from 15ms to 100ms and that is borderline unacceptable for a fast paced game.

In comparison pretty much every gaming monitor's selling point is 1ms, 2ms, 3ms and now even sub 1ms.


  [1]: https://www.rtings.com/tv/tests/inputs/input-lag