Yes that's how it works. The pressure switch (or prove switch) senses trhe vacuum created by the fan. It verifies that the fan actually came on and that your vent pipes aren't clogged. The entire fan housing can be removed but I'm not sure you can take that apart, so you'd have to clean it together.
However if it needed cleaning I would wonder why it works after jiggling wires to the pressure switch. Jiggling the wires shouldn't improve your venting so I'd honestly look at the switch itself. Those pressure switches can get stuck so that may be whats happening, then when you giggle it or pull a manual vacuum you break it free & it works until it gets stuck again.
As far as your saran wrap test I wouldn't judge too much based on that. Those motors only pull about 1-2" Water Column vacuum which is a very small amount of pressure. It may not hold something up to it. The only way to really test it is with a manometer.