Turns out I had simply forgotten how to pair my controller with a new device - I had to hold down the pairing button on the top of the controller, not simply turn it on. https://www.howtogeek.com/707503/how-to-put-your-xbox-controller-into-pairing-mode/
Thanks for being so patient. I’m glad the issue turned out to be so simple since that meant the solution was simple too.