Some fans are specifically designed to run by adjusting voltage (or, often, using a TRIAC control - but using a transformer / an autotransformer works better and reduces audible noise!) While far from optimal, this has been the most typical way of controlling fans, even fairly large ones (hundreds of watts)!
These are induction motor designs that have larger acceptable slip range and don't encounter breakdown torque so easily. The somewhat quadratic nature of the power required (as a function of RPM) by the rotating fan blades also helps here, torque requirement drops when you slow it down, so the unoptimal voltage control works well enough.
Generally speaking, the best and most elegant solution is to actually create a two-phase VFD inverter, since a capacitor run motor is nothing more than a two-phase motor, capacitor generates the second phase (which in theory would be 90 degrees out of phase). But tuning this to be optimal may be a lot of work. I have done this once and it was an interesting project, but makes little sense economically. The end result can be better than the original motor, however, since now you can exactly control the phase shift and the amplitude of the second phase. The capacitor is always a compromise.