I have an Xbox One S controller (Bluetooth) and a charge and play kit. Ever since I bought it, I've always noticed a slight burnt smell. There are no issues with the controller. No burnt chips, nothing getting hot enough to emit a smell.
I've removed the battery pack and confirmed that the battery isn't causing the smell.
The trigger rumble motors have always felt a little weak on the right trigger, so maybe it's a bad motor?
The controller works flawlessly minus a few repairs I've had to do to it over the years (I've had to replace the right stick 3 times now, the sticks in these controllers are so badly designed, my first repair was because the stalk snapped through normal use.)
Could a burning out rumble motor be causing the smell? Even if it's not in use? I'm honestly confused but just kinda live with it as the controller is otherwise fully functional.