Is that like a septic pump aerator? I was just looking. I guess its a diaphragm pump?? So I'm barking up the wrong tree with what all I was saying. That will act alot like a solenoid going in and out, working the diaphragm to pump air. I'm guessing thats what those coils are for on top of the pump?
EDIT...found it. Looks just like a Hiblow hp80 or so. There are many youtube videos on changing the diaphragms. I watched them and it looks as if you took the sides off and the diaphragms you can get to the spindle if thats what you call it that goes through the magnets. Maybe it has become dirty and is not able to move back and forth now. Still looking at stuff. Trying to find you a diagram for the motor part. It looks to be a very simple setup that shouldn't fail often. You said there was lots of dust. Maybe some electronic aerosol cleaner?? Maybe some dry lube or something on the moving parts.
Not any help, but trying to find you some information...its hot in the shop...lol.
Edit again:
You can get the elctromagnets online. I have attached links. The rod that goes through it is called a actuating rod. If it is seized up that would stop it from moving.
HP 80 parts actuating rod/magnet I would try to take the diaphragms off and move the rod back and forth and see if its binding. That may be why it starts then stops all the sudden. cleaning it all may help also.
Here is one of the youtube vids on taking the diaphragms apart. They do not get into the magnets, but you can see how it works once you get the diaphragms out. Pretty neat setup. Hope this helps.
Youtube vid of hiblow pump disassembly