Looks like classic symptoms of a fuel pressure issue.

Fuel pressure regulator bleeding down, rather than keeping fuel in the fuel rail, or a leaky injector doing the same thing intermittently. The fuel pump takes a few seconds to develop enough pressure in the system to inject enough fuel to run the motor. Could also be leaky check-valves in the pump itself, but probably not the pump.

There's a "tire valve" type nipple somewhere on most fuel systems, and you can connect a pressure gauge to it. The system should hold pressure for several minutes after turning the car off.

How the heck could your mechanic have not even thought of the fuel-pressure regulator, rather than throwing other parts at the issue?