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Hello! My name is Gary.Welcome to JustAnswer.I'm reviewing your question now, and will post back with a reply shortly.
Short answer, no.
All cars cut fuel on coasting throttle, so a shutdiwn interruption is no different.
Running out of fuel is also another scenario.
Sounds like a broken timing belt IS the cause of the shutdown.
Quite common for many makes of engines.
Pushing cars around in gear does not cause belt failure.
Belt replacement history on this engine?
If you have no belt replacement history, and so you don't know the age or mileae of the belt, then that it why it has ultimately failed.
All timing belts have a service life, they need to be replaced at regular intervals to prevent failure.
Belt failure generally ends up in bent valves in the cylinder head.
So in your case, the belt failure has shut down the engine, not immobilizer or alarm system.
Sorry to give you bad news here, but just agreeing with your shop diagnosis of belt failure.
Hope this helps.