Once started the car should run with the battery disconnected, but if the alternator was dodgy you would see the alternator light coming on, or the battery wouldnt charge and it wouldnt start.
Get a volt meter on the battery - it should show ~12v without the engine running, and ~13.5v running and charging. If it doesnt show more volts running then there is an issue with the alternator.
The problem you describe could just be a safety feature on the car, without a battery to dampen the current in the system, it may detect a spike, and kill the car so as not to do any damage anywhere else.
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