are you sure there isn't a blonw fuse for starter motor solenoid circuit.

Had a Peugeot expert that wouldn't start on key but would with a bump, checked for output at switch which was good, also had an input to the under bonnet fusebox.

I eventually found a blown fuse. This was caused by a lazy/faulty starter motor drawing to much current as i tested it with a clamp meter.

New starter and all ok.