Q: Idle air control stuck closed and the car won't hold at idle

Generally, a little air will pass through. But the thing is, the idle air control valve controls your idle speed. If that is shut, you are not going to get enough air bypass and the car will idle low enough to shut off.

There won’t be enough bypass into the throttle because the throttle is spring loaded, and it will always stay closed. You really need the idle air control valve to work right on most modern cars. They will stall out and shut completely when they do go bad.