I was explaining how it functioned per your question. don't take it apart. It is a major pain in the *** to try to reassemble. Buy another one and call it a day.
Model number ? Go look at any tractor supply that sells I/R for around a grand or under. IF you buy the 242,253 or 2420,2430,2520,2530,2475 pumps, and a couple other numbers in this range, then you have an I/R.
The switch sees air pressure from the bottom where it threads into the tank. As the pressure rises in the tank, it pushes against the diaphragm in the switch. the diaphragm is forced against the spring seat and pushes against the springs until the spring pressure is overcome. The contacts then...
AGREED, however, the less expensive IRs have a very cheap imported pump on them which IR only partially supports. I've had it apart and as stated, it is less than the quality people have come to know IR for.
Normally I would tell you the contact points were arced together be it pressure switch or magnetic starter or both, but in THIS case I would say somehow your pressure switch is either set too high or it is jammed and not allowing the points to kick out. if the former, adjust it down, otherwise...
You are right on in your thinking. Personally, I'd purchase COILHOSE filter and T-Handle regulator. They may not be the cheapest, but you have already traveled that road.