An update that I hope is helpful:
Thinking the system wouldn't let me standby because I had not actually landed yet, the moment the wheels touched the pavement in SEA, I tried once again on the AS app and it failed. I had no other choice but to proceed to the gate. The gate agent seemed as bewildered as I was, but put me on the SB list, and while there were 24+ already on it! I managed to end up at #6. The flight was pretty full as it was, so felt my chances were somewhat slim.

Seeking some clarity for future reference, I went over to the Service Center across from C2. The agent there was able to look into my record and she could see where I had tried 5 or 6 times to do the standby and failed. She scoured all of the elements of my reservation to see if there was anything tripping it up, and she could not find anything. In short, it should have been successful on my first attempt in DEN.

Conclusion: glitch in their system, to which nobody could override (the gate agent in DEN had tried that), except for the gate agent in SEA. One should be able to SDSB when checking in on the app anytime on the day of flight, except within 45 minutes or an hour of the flight once it switches over to airport control.

I was the last person who got a seat on the earlier flight, so all ended well.