Most small fans and blowers use either a shaded-pole or a Permanent-Split-Capacitor motor... both of which are in the induction motor family. (Some good reading about motor types
HERE, for those interested.)
Your router (along with your drill, blender, vacuum, car starter, etc) has a commutator-type motor, which is in the synchronous motor family (the other big motor family).
It is easy to vary the speed of a commutator-type motor just by varying the applied voltage... the commutator takes care of varying the electrical frequency actually applied to the windings.
Induction motors don't have it so easy. If you vary the applied voltage they *will* slow down, but if you don't change the applied frequency in proportion, they'll suffer from a sort of internal electromagnetic "slippage" (technical term is actually just that, 'slip'), which causes heating of the rotor.
This heating matters in big induction motors, like the ones in your washing machine, fridge, and air conditioner. (It
really matters in the thousand-horsepower motors you find in factories and mines.)
Fortunately, small induction motors have enough reserve cooling, and are already inefficient enough, that they can stand a little extra heat caused by high slip due to voltage-only speed control. "Small" is going to vary depending on motor, load, and installation... but in my experience, motors rated less than 100W, and especially fan motors (where the load automatically drops off as the speed reduces), are often suggested for voltage-only speed control.
I'm no expert, but if it was my fan, I think I'd try that router speed control on it, or perhaps a motor speed control from Grainger... after checking that the current ratings are compatible, of course. But I'd stop the fan every few minutes and apply the nose test to the motor (removing the blower from the stove if necessary). If anything smells 'hot'... either the hot-dust smell or the hot-paint smell... discontinue and run your fan on fixed speeds. If the motor shows no signs of distress, you're probably OK to go on running it at the reduced speed.