Light fixtures can build up a *lot* of heat and it is common for this to damage old wiring - the insulation crumbles when you move a wire as you learned!
If you have attic access to this fixture, you could maybe install a large junction box (plenty of room for connections) in the attic - say a foot back from the ceiling box, then cut off the damaged wiring, make the splices in the new junction box, then run a new wire from there to the ceiling fan box.
The old wire further back from the ceiling fan junction box should be ok since it would not have been exposed to the heat from a light fixture.
As to troubleshooting the fan. Sometimes it is simple. Did the switch on the fan get turned off? Or if that is not the problem, sometimes it is easier to troubleshoot if you remove the fan from the ceiling, then on the floor connect a length of wire with a plug on it to the fan wires, then plug it into an outlet and see if the fan works.
If it does not work, then take apart fan and inspect the wires inside and around the switch(s). Unlikely that the fan would have been fired or the light socket in the fan would have been fried, but wiring inside the fan may have come loose or been fried.