The tow switch appears to be working because my battery gauge doesn't work when it's on tow, so what do you make of the voltage on pins 3 and 4? Or should I be looking at 15 (or even 17)?
double check pins 3 and 4
they should be B+
if you are certain you are not getting B+ on 3 and 4
look at the wiring diagram you posted and begin tracing through the various connectors and the T/R switch, from the controller to the B+ connection
you are losing voltage in one of those connections
this is preventing the controller from turning on
it thinks the T/R switch is in tow
look for
loose or corroded connections, sometimes even the slightest difference in color is a clue
get a can of plastic safe electrical spray cleaner, use it liberally on all of the connections including the entire J1 plug into the controller
unplug and replug multiple times, this will help clean the connections
if there are no other problems fixing that will allow the controller to turn on and operate the cart
if there are other problems you will have to re-check all pin voltages again against the diagnostic you posted
IF your wire colors on the cart match the drawings wire colors (at least for the most part, sometimes one or several will be different), then you have the correct manual
the pics you posted from the repair manual as well as the complete manual itself are what you need to be working from and nothing else
anything else will muddy the water