This has most likely been mentioned before, but just in case it hasn't ....
The electric fuel pumps are wired (or supposed to be) through an oil
pressure switch. Once the engine starts, the fuel pump will only run if
there is oil pressure activating the switch. (the purpose is to ensure the
fuel pump doesn't keep pumping fuel into the carb if the engine dies or is
off. I believe that when you turn the key to the "start" position, the oil
pressure switch is bypassed to allow the pump to run and the engine to start
with no oil pressure.

From the symptoms you described above, it may be that the oil pressure
switch is bad or the wiring to and from it is bad.

Eisboch

I find it interesting that no one, except me, seems to have
achnowledged your point. Now that I have an electric fuel pump, that's
the 1st thing I'd check. Also, with my new ... geez I like my new boat
.... Volvo Penta, an alternator not working will cause the engine to cut
out. So many relays to jumper just to get back to port :-)