It's possible. After some research, it seems you are probably right about one of those things. I have not had a chance to test or replace anything since I posted this, so hopefully this weekend. I have a spare resistor already, and I bought enough wire (in the proper colors!) to replace all the wires. The only thing I don't have is a new ignition switch. I'm going to try jumpering straight from the battery + to the + side of the coil just to bypass everything and try starting it. I know that can burn the points, and that it won't shut off until I yank the jumper, but I will only try it for long enough to see if it turns over. Plus, I also have spare points lying around.
Thanks for the reply!