Thanks for the reply, Luidolt,
I am running the S22 Ultra (Android 13, January security update) as well, and I managed to get it working.
All I did was make sure I have the S Pen Air action toggle on (otherwise the event for the button press was not emitted) and restarted my device.
After the restart, I was able to receive the button event as well as the motion data from my S Pen.
Hope this will be helpful for someone.