If it is typical of other Fiats then there may be a non-return valve
not far from the pump. I find these can fail, often due to a build up
of  sludge or algae growth, well some sort of goo.
I'd unhook the outlet pipe from the pump and put a length of clear
screen washer tubing on the outlet, hold far end as high as possible,
run the pump to make sure it produces a good head.
That proves out the pump and that it is pumping, not just spinning
the motor.
I suspect that the route may be via a front door pillar alongside
the windscreen, following a similar route to the interior light
wiring.
With the outlet of the pump disconnected, blow down the
tube towards the rear screen to see if there is any resistance
or blockage.