While Bluetooth had something to do with it, I'm beginning to think there may be another issue with the way the controller is powered.
With the cable the power flow is constant. I have tried with the battery of a "play and charge" kit ( more or less two years old) and the experience has been negative: lag, glitchy, unresponsive... I have also tried with an old pair of rechargeable batteries (Apple) and the experience has been better but: I had to change the sensitivity of the sticks as the general feeling was to go a little slower, maybe a bit of lag. Compared to the cable, the game was more... heavy... After a while I had to go back to the cable because it could be decisive in a gunfight... :)
Yesterday I was playing fine with the batteries that came with the controller. But after a while I got the low battery warning and became a bit slow again. I used a tester and the voltage level was 1.45 of 1.5, so they still have quite a lot of life left.
With that so many different sensations in different scenarios, I would say that either my controller is defective (and that may be the reason why several users report that it works well for them) or it needs a firmware update to better manage the battery charge, perhaps related to the bluetooth LE.
Of course it's just my perception, but let's see if any other user can practice this kind of tests (with intel bluetooth driver v21.110.0.3), to request a controller replacement or wait for some kind of update from Microsoft.