Timing is too far advanced. That is the symtom. Loosen the hold down after it's been driven and is hot. Bump it in the retarding direction slightly. Try to start. Continue doing until you can just bump the key to start. Lock it down. Works every time.
retarding the timing will lessen the load on the starter, but just because it starts better there doesn't mean it's the correct setting... the correct setting is where the car runs best at, which might be where there is too much pressure to start well with the starter.
If that's the case, make sure the battery, starter and it's wiring is good first (they can get weaker as they get older), you might want to replace it with a high torque starter (not sure if one is available for a 351C, you should be able to find one for a W), if that's all fine then you might want to consider either adding an ignition amplifier box that has a start retard feature (many of the newer ones have one automatically built in), or wire a switch inline with the ignition circuit that allows you to crank the engine without the ignition firing, then you can start the engine cranking and engage the ignition once its turning well. Really, this shouldn't be necessary on a street car, but some of these were built when high compression engines were still common and were built for higher octane gas than is easily available now, and heck, they're 40 years old, who knows what someone built and dropped in in the last 40 years.