Is the fan maybe humidity controlled? Any small dials / knobs that are visible on it's back? We had such a bathroom fan, and the humidity detector was faulty, but luckily, twisting the "sensitivity dial" to max, just kept the fan on continuously.

Other than that, does it spin if you just push it? As in; is it able to rotate freely?

Might also be nice to know what make and model it is if that's available =)

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One thing that stood out, was that you measured only .3 volts at the switch? as in 300mV?.. That's absolutely nothing, so either there is no available power there, or something on that line is short-circuited (which would drop the potential to nearly 0 volt) but that would also cause any fuses/circuit breakers to pop.