I just tried it again. I first turned it off, while leaving the PS turned on. I then tried just pressing the on button, and holding it for a second (I counted), and it doesn't come on. I then pressed and held it for a count of two seconds, and it came right up.
If I leave the PS off for more than a count of ~10 seconds, turn it back on, and try turning the radio on, I can hold the power button down for longer than the 2 second count, and it doesn't come on. If I then release the power button, and then press it again, the radio will come on after holding the button down for 2 seconds.
I just thought of something. It might have to do with the radio's power supply stabilizing before it powers up! I just tested that, and it appears I may be on to something. I first turned everything off. I waited about 30 seconds, and then turned on the PS. I then waited another 15 seconds, and then tried turning on the radio, and it came on after a 2 second press of the power switch!
So, that must be the issue, as I have experienced this with other equipment in the past, where you have to first let things settle down a bit before attempting to turn them back on if their power source had been interrupted. It sounds like their SDR circuits need the internal power supply to be completely stable before it attempts booting up. It probably puts the radio in a stand-by mode when you turn it off, leaving it's power supply energized and stable so it can power right back on when you hit the switch.
Thanks for the help!