My light won’t come on

The light to what exactly won't turn on?

That would be the LED as shown in the manual: #2 LED Indicator -The LED indicator provides system alerts and feedback for different operating modes.

I have the same situation. I have an Xbox Series X and it does not light up green. Can the KeyMander operate without it being connected to the app fulltime?

Yes, it can. You only need to connect to the app if you need to change game profiles or settings for your profiles. Otherwise, the K2 will still be operational on the last loaded profile even when it is not connected to the app.