Well, my first thought as to why it won't start is because the carburetor is off of it! LOL
Seriously though, need more information to know just what is going on.
Is this a magneto or battery ignition?
The symptom of miss fire under load, quitting, then restarting, could be either ignition or fuel related.
A simple test is to try a partial choke when it begins to misfire. If the performance improves, the mixture is too lean. If the problem gets worse, or there is black smoke before choking, the mix is too rich. If the problem remains unchanged, the cause is likely weak ignition.
As for the Carter carb, those weren't very popular carbs. They were troublesome, and from what I understand, it's hard to find parts for them or make them work well. If you are having problems with it, might be a good time to upgrade to a Zenith or Marvel, which can be bought new from YT.
If you do replace the carb, be sure to hold on to the old one to be sure you get the throttle and choke linkage back the way it was. Especially the throttle linkage, it must be right for the governor to function.
Also check the quantity and quality of the fuel flow. Before connecting the fuel line, open the fuel valve and catch the gas in a clean glass. You should get a full flow, not a trickle or drip. Also check for rust flakes, dirt, water... You don't want that going into the carb!
Back to the ignition, if this is battery ignition, when you converted to alternator, did you install a ballast resistor ahead of the coil? If it still has the 6 volt coil, it could be overheating and loosing spark. Even the proper coil can fail, it will give problems when hot. Also check the condition of the distributor shaft bushing, the point gap and condition, wires, cap, rotor, plugs...
If magneto, they too can fail when hot. The old coil can be failing from the insulation drying out and cracking. They also have points that need to be kept set and clean.
Either case, check the spark at the plug when it fails. Need a strong spark that will jump 1/4 inch.
As for the broken alternator belt, no it won't prevent it from starting. If there is enough battery charge left to crank the engine over normally, there is enough to power the ignition. If magneto, it will have no effect at all. But, you can not run it this way! Without the alternator belt, the water pump and fan are not turning!
