Well, it has been a long time since I worked on them on small engines. Here is a nice page if you want to study it: http://www.simpletractors.com/service/e ... erator.htm
I'm not entirely sure but looks like the R and G on the starter is likely the field coil, it controls the charging. The regulator grounds the field to let it charge and cuts the ground many times a second to slow charging when battery reaches full voltage (13.5-14v). That is why you should be able to see what it is doing with a meter on the battery. Battery would be 12-13 at rest and above 13 running.
It looks like B is ground, regulator must be grounded to control the charging. You did check the fuses right? They like to blow sometimes. I can't remember if the starter shuts off or not. It looks like R is hot and the foot pedal makes R/W hot to run ignition coil, voltage reg, ignitor, and starter. The only difference here is the old generators run the field right off the large starter winding and this is separate. Since the starter relay and ignition coil run on the same hot wire I don't see how it could be shut off after starting. Unless the starter does it on its own I'm not sure. I know if you let off the pedal at speed it does stop charging.
Some generators run differently, with ground or hot on field. I know the GM and Ford cars from the 50s were not the same. I would also put a meter on it and check wires for poor connections, see if it starts charging or has bad ground...something like that.