the same thing, the car leaved me out in a road once, though I was low in fuel, by the time I get some I started fine; at home garage once, reached the same conclusion similar solution; but the last time, it leaved me out in a turnpike with more than 3/4 of a tank, hook the car home and decided not to go out until solved. Took the fuel pump out of the tank and removed it from the plastic assembly, when to the jump start post, and start connecting to battery voltage and disconnecting it. Noticed not every time it started running, sometimes, having good 12 volts on its terminals, it didn't start, so decided to get a new one. It has been about 5 years since, no problems at all.
The pump will no run with key in run unless engine running, best check is to jump relay, as you did.
Conclusion, yes the fuel pumps some times say goodbye reluctantly.
Javier