This button normally flashes and then stays lit whenever the controller is on.
That makes the Xbox button light the best indicator of whether or not one of these controllers is working.
Start by removing the battery compartment from the rear of the controller, then remove each battery.

Jeremy Laukkonen
If installed wrong, put them back in the correct way.
Otherwise, try afresh pair of AA batteries.
Just because batteries work in one thing doesnt mean they have enough power for another.

Check the battery contacts.
Remove the batteries before prying, and be gentle if you do attempt this fix.
Prying too hard may break the tabs.
Test your Xbox Series X or S Play and Charge Kit.
Try them with a different controller if possible, and see if it turns on.
If it doesnt, then either the cable or battery are bad.
revamp your controller firmware.
Xbox Series X or S controllers are complicated devices that require built-infirmwareto operate properly.
If an update is interrupted or the firmware is corrupt, the controller may not turn on.
Connect the controller to your Xbox using aUSB C cable, and see if it works like that.
If it does, consider using the controller as a wired controller.
It probably has an internal fault that prevents the controller from working on batteries.
Contact Xbox customer support.