Think of it this way:

Everything runs off the battery. The alternator charges the battery.

If you have a dead battery it doesn't hold charge, but allows charge being put into it by the alternator to go straight through to the devices using power. This is fine so long as the engine is running. However it's tough on the alternator to feed a draining battery plus all accessories all the time. This can lead to a dead alternator.

If you have just a dead alternator then everything runs off the battery until it is depleted. As in you're driving home at night and the headlights get dimmer and dimmer and dimmer... until dark. You will know a dead alternator soon because you will kill the battery in short journey using power accessories at night.

But as you've discovered in this thread, the diesel engine will not stop for lack of battery, or alternator, once it is spinning. You can (and must) stop the engine by starving fuel. Vacuum is used to starve fuel when you turn the key off. There is a vacuum circuit through the key cylinder, running in parallel with an electrical circuit going through the key cylinder. This remote stop feature does exactly the same thing as you would do by pressing the STOP level on the IP under the hood.

Once stopped, you will not start with dead battery. Getting a boost can get your car running. Or, you can also push start the car, even though it is an automatic (sometimes a little known fact about MB's). Now it's running again in the dead battery condition (above) and no power accessories will work.

It's not fair to say "your car works fine without a battery", but it is fair to say "your engine will run without a battery". You'll agree driving at night in a rainstorm is impossible without good voltage in the car.

__________________
Cheers!
Scott McPhee

1987 300D