It's probably just anecdotal that the air pump turning off coincides with the improved idle. All the puump does is blow a little air into the exhaust manifold to help heat the converter. Of course, the pump might be failing and creating enough drag to pull the engine down; but that's very unlikely, the engine could shred that small pump at idle and never even notice it.
To quickly check the pump, just unplug the wire going to the clutch in the front of the pump; it will then run on the idler bearing like the AC compressor. You can plug/unplug it and listen to how it runs and tell if there's an issue. If there is, just leave it unplugged, engine will not care.

Your problem may be in the cold-start valve - but I'm not an expert on that motor, let's see what Doc says.

DG