Can someone explain why the modern car radio, even in a Prius, momentarily shuts off when the car is started? Is it taking up that much current that it had to be disconnected? Does it happen to AC too?
Yes, because the starter motor draws so much current (up as high as 200 Amps for some cars), any non-essential electrical devices are usually shut off until the engine starts, to ensure the starter can do its job.
You might think it doesn't make a difference, but in reality, starting a modern engine really takes it out of a battery. If you ever want to test this, see how fast you drain your battery trying to start an engine with no fuel. It doesn't take many attempts...
It isn't a deliberate design feature. It's just a consequence of the fact that when the starter begins to crank the car, the battery voltage will dip significantly, maybe as low as 9 or 10 volts instead of 12. The car radio circuits (which these days are mostly digital signal processing done by computer-type circuitry, not the "simple" analog electronic circuits you would find in a 1960s or 1970s vintage radio) can't handle that, and the audio output gets interrupted.
It is easy to "see" the voltage drop if you start the car at night with the lights on. The headlights will go dim while the starter is cranking.
If you really want to pay a few hundred dollars more for ICE that doesn't do this, you can probably buy it - but it's not a cost effective option for standard car audio equipment.
Incidentally, the same thing would also affect the car's ECU computer, except that it is designed to deal with the voltage drop - if it couldn't, the car would be impossible to start! Before electronic ignition, there was a mechanical switch connected to the starter switch which effectively set the ignition circuits to work on 6V not 12V while the starter was cranking - and that had the side effect that if you left the ignition for a long time on with the engine stalled, you could fry the ignition electrical system by overheating it with 12V when it was only expecting 6!