Nevermind, I got it...
Seems that the console requires that at least one SENSOR (not remote) be installed at the console (not in AHP) in order to arm/disarm.
If I start from scratch and install only the remotes at the console and in AHP, then it does not work. I can hit whatever button in OnAlert Mobile and it does generate the code and does show it in the log (and I suspect actually is sent via RF), but there's no effect at the console. As soon as I install a sensor (e.g., DS10A, MS10, etc.) at the console, then OnAlert Mobile starts to work properly (even without yet adding the sensor in AHP). Which all makes sense.
Anyway, not something that anybody is likely to run into other than if they just happen to try it in the process of setting things up. I had cleared everything at the console and in AHP and was again setting up AHP, OnAlert, and ActivePhone, and the remotes, but had not yet gone back to re-install the sensors.