There's a relay - a big switch - that turns the electric fuel pump ON
and OFF. If the relay opens the fuel pump stops and the motor stops,
period. Your car's fuel pump relay is suspect - due to these very
symptoms and its age - they get tired.
That a professional repair shop would be mystified by this problem is
unbelievable, we see this here weekly! They're trying to pick your
pocket, IMHO, and my compliments to you for resisting saying "fix it" -
regardless of cost.
The relay isn't cheap but can be bought on-line from various parts
suppliers. It just plugs in.