You have already proved the other side of the wiring is fine with your jumper because your pump fires and runs and then stops after a while (presumably until it reaches a required charge pressure for the system).
Not the case all the prerequisite's are bypassed and the pump is just direct powered.
I "assumed" the bridge wire came out a bit and lost contact.
Its the control side needs checking first. now if the pump was just running and not producing pressure it would just continue to run (may trip on high temp).
When my pump runs I can hear the change in noise as it gets up to pressure, so it could still be the pump as his brakes seem weak.
J