Yes, that's what happens electronically. However, there seems to be a sequence thing and I'm not sure exactly what it is, perhaps opening the door with the engine running or an auto trans not in park where the radio does stay on for a bit after locking the car. I've noticed this a few times with several different VF's at random occasions.
not having it in park prevents the shutdown message being sent.
on some gm vehicles you could shut the engine off while in park, open the door and then shift to park to keep the system running.
with my first colorado i had a switch on the swcan wire to the mylink that i could use to keep the system on and play video while driving by disconnecting the swcan wire with the switch.
if you power on the mylink without the canbus active it will shutdown after 10 mins.
normal operation of the vf is once in park and engine off the shutdown message is sent when the drivers door is opened, this starts the sequence of all of the modules going into sleep mode.
i think more of these gremlins will start to appear as these vehicles age because the manufacturer never really designed the electronics for longevity, some of the modules i have tinkered with i have noticed they write to non volatile memory on shutdown/sleep and this type of memory has a write life so only so many writes can be performed until corruption and that will cause failures.