I know you said that you checked the grounds, but this sure sounds like an intermittent connection, ie: poor ground.
Those other heavy gauge wires you see at the starter are the (+) wires that feed the ignition switch, fuse box, headlights, etc. They should be fuseable links.
If the wire colors are correct, the purple wire is the start wire, connected to the "s" terminal of the starter, and back to the Ign. switch.
If you can catch it in the act, do a voltage drop from the purple wire at the starter while cranking. It should read extremly close to zero, and not more than around .6 volts. (FYI- it should read 0 volts key on in the run position) If it reads more than .6, you have a bad connection (corrosion) or a poor ground.