I actually managed to find some notes and example schematic of a driving circuit in an old catalogue PDF from the manufacturer of the fan (Papst). I now know that I need to feed it a 1-6kHz PWM signal using an open-collector output, so no need right now to bother stripping any of the conformal coating in order to go poking at the PCB.

I don't have any of the suggested stuff on hand, but as acetone is the nearest thing to lacquer thinner, I still might try out using some on a non-important area of the board, just for an experiment.
Regarding fault-finding on the fan, even though I can now operate it on the bench at variable speeds, I think something is indeed faulty in the fan's circuitry, because it won't
stay running. It cuts off after a random period of time - anywhere from a few seconds to a minute. Don't think it's an overheating problem, as none of the ICs get anywhere near warm. Cycling the power always gets it going again immediately, which is odd. If I can be bothered, further troubleshooting will be dependent on my success with experimenting with some acetone to strip the lacquer.