I played competitive FPS games for years at high ranks with a Bluetooth headset before learning of the delay. I did not notice. However, for a while I switched to a Bluetooth keyboard and until I realized the input delay was severely impacting my performance. Wireless mice (non-Bluetooth) on the other hand have low enough latency that most pros are fine using them.

Some advice for Bluetooth headset gamers - Bluetooth completely breaks the audio for some games like Battlefield 1. The only solution is to go into device manager and disabling Handsfree Telephony on your device.

While off-topic, somehow Apple has screwed this up in the M1 MacBook Air built-in audio hardware too. I wanted some monitoring while playing acoustic the other day so I opened logic and rather than connect my expensive audio interface, I just wanted to use the built-in mic and headphone output to monitor myself playing and singing in real time. For some reason there was still an audible delay even though I wasn’t using plugins and had the buffer set very low. I’m pretty sure it was in low latency mode… you shouldn’t have to use an external audio interface just to do real time monitoring like that such a simple use case.