I do not have an Escape wiring schematic so I can only theorize but the odds of a new pump being bad are about zero.
Seeing as how the pump does not operate when the key is operated at the time the problem occurs I would suspect one of the following, in order of priority.
Faulty trigger circuit for the pump relay. (Relay is ground through the PCM for a few seconds and this ground is maintained if the starter motor then cranks the engine.) Wire connector scale/corrosion, faulty PCM, etc.)
Burned contact points inside the pump relay.
Scale/corrosion on a fusible link connector in the underhood fuse block.
I’m assuming here there are no codes present?