I had to Google it to see what sort of fuel pump your car has .. I'm new to Fiats..
You've got a mechanical pump, yes? Remove the pump and use a torch/mirror if need be to look inside .. you're looking for wear (usually a groove) in the cam/lobe/ring that actually operates the fuel pump arm.
The symptoms you describe fit a worn cam/lobe/ring. The arm of the fuel pump wears a groove in it, and eventually, because of the wear, won't operate.
You fit a new fuel pump and the car runs ok - because that pump is new and the tip of the arm is 'bedding in' to the groove. A few miles down the road, your car cuts out again ..
It was a very common problem with mechanical pumps. Removing the gasket between pump and block is a temporary get you home measure (it allows the pump arm to reach just a little further..). Another trick way back was to braze a dollop of metal onto the tip of the pump arm - so it could compensate for the worn groove .. but again, this was a temporary measure that only lasted a few weeks.
You can fit an electrical fuel pump somewhere in the system - there's no need to remove the old mechanical pump as the electric pump will push fuel through it.