OK, heating the switchbox is crap. If that is the cure, you have a switchbox issue. But hold off replacing it, until we test further.
Let's start with the basics. The starting procedure is as follows. Pump primer to fill carbs. Stop when primer is hard. Lift cold start lever all the way. Hold choke button and crank motor until she starts. Feather choke and cold start lever until she is warm.
If that doesn't work, check to make sure the two carb choke shutters are closing when choke button is pushed. Put a timing light and the cold motor and crank it to see if you have spark.
Post back.