Sorry for replying so late everyone, but I waited on the new sensor for too long and then forgot to come here and give an update once I replaced it.
So I put the new oil temperature sensor in, did my usual to/from work rides for a couple of days and still couldn't hear the fan. I rode some curvy mountain roads over the weekend however, and once I got to the top of the mountain there it was in all its fan glory - it was working!
Now the only "problem" I have is that I still don't know if the old sensor actually broke or I just didn't get to hear it at the times I was paying attention to it. I should be able to heat it up to 140C and check if the circuit opens (the sensor is specified to 130C +/- 5C).