if i were you, i would be taking the board out, and carefully cleaning it up!
scrub it down and see if i could identify from the circuit board what the components are or rather were.
be very careful as you clean, so that any printing on components doesn't get wiped out, or scrubbed off. use a small soft brissel pain brush to start, take a picture, blow it up and then mark on the picture any identification numbers on components you can find, and any marking on the circuit board that might help identify the part(s).
my bet is this is the voltage regulator and that is why you are down on voltage.
you also want to carefully clean the brushes, the brush holders, and using very fine wet or dry paper the slip rings on the rotor, make them bright and shiny again.
i don't have any schematics, for your genny, or that pc board, but with some cleaning, some better pictures and as much info from the parts and the board, we ought to determine what parts are and what is needed to get it functioning again.
the genny is not dead, it produces power, so in my opinion, the parts that are burnt are likely going to be fairly easy to determine what they are and their values.
good look
bob g