The question I've got is: is there any reason why I shouldn't pull thegizmo
that's there, and replace it with a conventional toggle switch???

Since the switch and fan assembly were designed to work together it would be
prudent to keep things original or else replace both fan and controls with
modern units. Might be the heat you're sensing is from an early type of SCR
dimmer or speed control in the center section. Whatever, proceed carefully and
use well rated components. Good luck.

Joe