Update: able to reproduce the problem and come up with a fairly consistent fix. Which sort of points to the problem already (surprised the devs haven't figured this out?)
Fix (seems to work for me - need more testing to see how consistent)
1. Restart of console (not "turn off console". It seems that the XB Series X "turn off console" is more of a sleep mode unless you actually pull the power plug)
2. Start NHL 23
3. In the game, when it comes to the controller selection, resync the controller (press the connect button on the console and controller).
4. After resyncing, your controller is finally the red controller.
I did this last year for NHL 22 and it seemed to work. So I'm pretty certain it's the devs reusing code.
EA, please make it a priority to patch the game for the controller issue! It's been over a year already!
Edit: for Devs, I have the latest game updates, latest firmware, latest console updates. Everything's been updated. Every other game works fine EXCEPT NHL 22 and NHL 23. No problem on other EA games using Frostbite either (e.g. Battlefield 2042) - I don't have FIFA or Madden so I can't check either.