I’d suspect a problem with the carb. Who rebuilt it? There are settings for float level, idling mixture, and perhaps running mixture. What the settings are and where the screws are differs depending on the make model and size of the carb you are running.
If it idles smoothly but isn’t getting gas to run when you press the gas, you can check to see if the accelerator pump is connected properly and shooting a squirt of gas when the throttle is pressed. If yes, move on to checking the float levels. Not enough gas in the bowls could cause the carb to lean out when you give it gas and stall the motor.
If you get black smoke when you restart after stalling, perhaps you have a backward problem of too much gas when you open the throttle. Carbs are tricky. Something not installed properly in the rebuilt, adjustments that are off, and dirt somewhere can all cause problems.
The distributor should also have a vacuum advance to advance the timing very quickly when you hit the gas petal. If the vacuum advance isn’t working usually the motor bogs down for a bit until the centrifical advance kicks the timing up. So, a distributor problem could be part of your problems too.
If you squeeze on the gas very slowly will the motor pick up rpm’s? Any backfires?