I've been digesting the failed compressor a bit more, after having a better grasp of check valves, exhaust valves, and unloader valves and trying to make a map of this particular system in my head.
I think what's failed is the exhaust valve in the pressure switch. The machine isn't over-pressuring.
Order of operations of this machine:
1. At empty pressure, machine is turned on by operator. Pump begins cycling.
2. Check valve between pump and tank allows positive pressure from piston to be input into tank without backfeeding into cylinder.
3. There is no unloader valve between/on the check valve and the pressure switch. This was 50% of my blockage in understanding it, since most serviceable compressors have an unloader valve to prolong pump life.
4. Air is stored in cylinder, and accessible via the manifold on the output hole in the tank.
5. Manifold has a dial for tank pressure, a nipple for a 1/4" nylon tube that connects back to the pressure switch and the exhaust valve, and a regulator with a quick release tool attachment.
I was interpreting the nipple on the manifold as the unloader valve. I was about to buy a new check valve, but I stopped myself.
The pressure switch has a metal plate with a receiving nipple for that 1/4" nylon tube, a rubber diaphragm, and that screws onto the plastic frame of the pressure switch assembly. But, there's a hole under the diaphragm and some sort of solenoid operates in that hole. I suspect this is the exhaust valve, a safety feature to bleed pressure if too high a pressure is attained.
What I'm hearing is the pump hitting target shut-down pressure, and the failed exhaust valve squealing and letting air out until the machine hits kick-on pressure and it starts filling again.
The handul of sites out there that carry specific parts for this compressor insist that if you hear hissing/noises from the side of the pressure switch, then the culprit is the check valve. But the problem is, their diagrams demonstrate that the check valve has an unloader line to the exhaust on the pressure switch. On mine, the manifold connects to the pressure switch and the check valve has no T-nipple. There's no way for the check valve to cause the problem here.
I almost ordered a new check valve today off Amazon since I was thinking that the problem was a leak from main tank into compressor pump, and the squealing culprit was the unloader valve getting too much air from the tank. But after looking more carefully at the pneumatic circuit of this particular device, there is no unloader valve and the air pressure that causes the squeal is coming directly from the tank via the output manifold and has nothing to do with the check valve.
I suspect, in order to fix this, I have to order a new OEM pressure switch.
Or, alternatively, I order a generic pressure switch, retrofit/kitbash a bunch of NPT fittings, use an aftermarket generic check valve with an unloader since the generic pressure switches all have unloader fittings, and block off the nipple on the manifold that is currently allowing the failed pressure switch to vent via its exhaust valve.
I'm not keen on spending $40+ on this while in doubt about the right step though.