It turns out the issue was between the controller and the chair.

I'm posting my resolution here in case someone else ends up with the same issue and goes looking...

I miswired the controller (it came with no documentation), but I came across this blog post (https://cy-822b.blogspot.com/2016/08/the-cy-822b-usb-led-joystick-controller.html) that showed this image:
c0fb63ff-f849-4c20-b512-ae2dd514b54c-image.png

The controller board looks like:
856a91fc-99f3-4ced-b08c-acd40ff39054-image.png

The buttons have a 2-pin connector on them so I figured the connected to the lower edge of the image above. But the connection actually happens in the button housing and the 3-pin connector is more than just for the LED, it also contains the switch wiring. So the 2-pin connector connects into the button housing and the 3-pin wire from the housing connects into the controller board.

I went back to the original Amazon page and found that there are some pictures that describe the wiring, but not terribly clearly:
cd4a53c7-a851-42b4-875b-919802fd6439-image.png

In the end all on me. I hope this helps someone else in the future.

Cheers,

Steve