Can fitting an extraction fan increase the amount of dust drawn into the cabinet?

Yes.

An extraction fan removes air from the cabinet, which means replacement air has to enter through the other openings. Any dust suspended in that incoming air is invited along for the journey.

The fan doesn’t manufacture dust; it increases the amount of air moving through the enclosure.

That is also why fitting a brush insert doesn’t make the cabinet dustproof. The cabinet remains deliberately ventilated.