In most other applications electric fans have a capacitor for starting up, and some have a second "running" capacitor. Looking at that picture I noticed the yellow thing under the capacitor, and it made me wonder if there could be some sort of safety plug that pushes out if it is blown. Unlike metallic versions with areas like a jar lid's tamper evident depression that pop up usually plastic capacitors just look obviously fried with what looks like molten lead bubbles protruding through melted spots, but automotive parts might include some more tidy precaution. You can discharge the capacitor somehow and test resistance to see if it's good. The reading should start low and gradually increase to infinity. If it stays low or begins at infinity it's trashed. EDIT: I see you said that's another terminal, but it's still a good idea to check the function because a faulty startup capacitor would explain why a "known good" fan is working sporadically. If you short the capacitor to discharge and don't get a healthy spark, you'll have instant confirmation, and I assume it could lose its ability to hold a charge without looking totally scorched.
