PSA: connecting a wireless XBox 360 controller to your PC with the USB cable does not in fact make it a wired controller. That cable is called a “play & charge” cable. All it does is provide power to the controller but does not transmit any data. Therefore, a wireless receiver is still required.
Illogical, but it’s the truth. When you plug it into your PC, Windows will complain that it doesn’t recognize the device and that it may be damaged, but nothing is amiss. Just a stupid decision on Microsoft’s part.
Yes, I really did waste too much time on that.