I would guess that there is something obstructing the ejector nozzle. Shutting the pump off reverses the pressure against the footvalve. This action may be dislodging the obstruction (at least temporarily).
The nozzle is installed right before the impeller. When it's plugged up, the impeller doesn't get enough water, and the pump won't build pressure.
There's nothing wrong with the pressure switch, although you should replace the line between the pump and switch. It seems like once they break, they'll continue to do so.
For an exploded view of a shallow well jetpump go here:
You'll be able to see the nozzle.