2008 Mazda 3 engine cooling fan won't work

Approach my electrical accessory advice cautiously because I've spent 4 years as a Yamaha pwc tech chasing mysterious issues in jetskis and boats caused by saltwater, and/or overzealous use of various protective sprays on wiring harnesses. I sort of assume every issue will likely waste countless hours of my time and immediately throw caution to the wind. That being said; I would take a battery and attach wires with to the terminals and clips or pins to them to check the fan's function, if it works I'd start at the wiring closest to the fan and carb clean the hell out of the wiring and connectors, then blow them all out with compressed air (I know, it's basically the original Nintendo troubleshooting method lol). If that doesn't help the issue or lead to the discovery of any damaged or corroded connections, I'd then replace the pins in the relevant connectors for the fan. It could all be for nothing, and I know that's not particularly helpful, but atleast then you'll know that you're not going to chase complex theoretical problems and replace costly non refundable components just to find out you had some faulty connections giving intermittent readings. Honestly I might be inclined to go to a salvage yard and cut that length of wiring harness out of another car. I've had a wiring harness that looked factory fresh, but after testing everything (almost literally) and finding no faulty components, I swapped in the old ragged, corroded harness from a totaled vehicle and the revolving playlist of error codes and intermittent symptoms were gone. It's almost never the ecu/ecm, but people (including professionals) always seem to jump to that conclusion when the electrical problem is hard to pinpoint. Good luck

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.

The oscilloscope diagram shows a square wave having about 14 volt amplitude (5V/division), 5ms period (1ms/division) which is a frequency of 200Hz, and about 55% duty cycle. What you will measure on a multimeter depends somewhat on the specific meter used. On DC voltage measurement you would expect to read the average (the DC offset), around 7.7 V. The value displayed might be lower, though. On AC voltage setting, the meter should only display the AC part, ignoring the DC offset. So a good "True RMS" meter should read near 7 V. A more basic meter that averages the voltage and scales it assuming it's a sine wave might read around 7.6 V.

If your fan control module is only putting out a few millivolts, there's a chance it has died, or maybe lost power, or is not being told to turn on the fan. But the fact that the fan does come on intermittently suggests that there is a bad connection somewhere.

The component pictured is indeed a capacitor (aka condenser). It looks like a film type. The "M" in the box looks like the logo of Matsushita (Panasonic). Capacitors have only 2 terminals; I wonder if the metal tab is just for mounting. Judging from its value, 0.47uF, and the fact that one lead is grounded, I would assume it is there for electrical noise suppression.