This probably isn't the culprit, but a bad wire at your ignition switch can cause all kinds of headaches. I warn people of this because I've been put through the ringer with this one.
Engine will not crank
Low battery power. Charging system while under way not sufficient to charge battery, but when at the dock the battery charger will raise voltage up to the point where it will crank and start.
Loose wire in the starting circuit. I know it sounds elementary, but the battery connection, the wire leading to the battery connection, and the other end of the battery where it bolts to the starter all need to be checked.
Along the lines of the above, check the ground circuit, that negative side of the system. Be sure the starter is well grounded.
Defective solenoid can cause starting issues, but not under the symptoms you describe, and it has nothing to do with cutting out. I list it here just as a checklist item.
Insufficient battery size cable, or old cable that has corroded internally to the point of no longer being able to carry the load.
Potential problem at your cut off switch. If the switch is defective, or has a loose connection, it will act just like a bad connection at the battery itself.
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Engine cranks but won't start
No fuel
No spark or weak spark.
Distributor cap brush (or rotor) is broken. Since this is used with the Pertronix, it is to be checked.
No spark at coil high tension lead.
Flooded engine
Sticking choke plate. Check thermostatic spring housing adjustment. On mine, I have deactivated the chokes all together because of fear of malfunction with 40 year old CArters. I just prime the dickens out of the throttles prior to hitting the ignition.
Fuel not reaching the carb, clogged shutoff valve or pick up tube in the tank. Try running with a temporary outboard motor fuel tank, which diagnoses this one IMMEDIATELY.
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Engine starts, but fails to keep running
Idle fuel mix improperly adjusted (or clogged with debris).
Choke not operating properly (as noted before).
Carb FLOAT setting not set properly (anyone been inside those carbs lately??)
Fuel pump defective (they do wear out eventually). What might look like a defective pump may be a clogged pick up tube.
Dirt or water in the fuel lines, filter, carb, or your fuel tank.
A bad coil will allow the motor to start, will overheat, and will cut out and won't allow re-starting until it cools down.
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SUMMARY
When the motor cuts out, if you are in a safe place to do so, I would immediately open the flame arrestor and have someone prime the throttles. If you have fuel in the carb bowl, you will see the accelerator pump squirt gas into the carb (this is one giant step in ruling out the clogged pickup tube or faulty fuel pump theory). Certainly if you have fuel in the bowl you don't have a starved motor.
It sounds like a poor electrical issue to me, because of the hard starting. I would check the grounds from the positive end of the battery follow it to every single step of the way, be sure everything is tight and as new. Do the same with the negative terminal, every step of the way. Just curious, do you have a battery switch to combine two batteries together for starting purposes? Your chrome button at the helm may do this via a solenoid mounted under the console, as mine has. Otherwise you could manually combine batteries for testing.
Good luck! Hopefully some of the other guys will offer some thoughts too.