How do electric propeller governors (or electrically controlled governors) work and what are their pros and cons?
They are much more prevalent than you think, and have been in use for more than 70 years. Rotax 912 and 914 engines can be installed with electric governors from MT Propeller. Airmaster sells them as well. You may also be surprised to learn that nearly 90% of German operational Luftwaffe aircraft during WW2 used an electrically-actuated propeller system designed by Dr. Hans Ebert of the VDM (Vereinigte Deutsche Metallwerke, or United German Metalworks). One example of these excellent propeller systems is in the Smithsonian's National Air and Space Museum.
This article does a pretty good job of explaining how they work.