Hello All,
I've breadboarded Ian.M's circuit and it works way,
way better than what I had before. Fan's speed makes a linear, steady progression throughout pot rotation. Thanks everyone for your help!

I've checked the PWM pin signal without the fan connected and it is a crisp square wave which goes from almost zero to 100% duty cycle. Frequency tends to drift a little along the pot travel, though, and seems just a bit on the low side, ranging from about 19 to 22 kHz.
Hencewhy I wish to trace the output of the tach pin, to check if the fan goes the whole range of RPMs (from 500 to 2000, roughly). I did connect the oscilloscope probe to the tach pin, but I don't understand what I see. It looks like a garbled version of the PWM signal, which makes no sense, because the tach signal should change in frequency but not in duty cycle (actually I have not read this anywhere but it seems obvious enough).
Where do I have to stick the oscilloscope probe and ground clip to properly read the tach pin signal?