The fuel pump is sucking gas right where he has the can, evidenced by the pump test. He is doing it correctly. If cap is off gas can for hose to enter, it's vented. Don't need a primer bulb either. This is good, eliminates all the worry about particles, almost. It eliminates vacuum lock, bad gas....all good news. Hope it's not tractor diesel.

The squirt down carb is not super safe. It is an extension of what is done to get a motor started. Yes we are trying to completely bypass the carb and the pump for long enough to get information. The motor does not know where it is getting the gas charge. Probably should NOT do the squirt while running if you feel uncomfortable. It is pointing to the carb now anyway if everyone is OK with the electricals. What new electrical component can break down after one minute and rejuvenate and do it over and over?.. and completely stop all spark? I don't know. It's not the symptom of a bad condenser.
It still starts and runs for a minute or so. That is good. Something wrong inside carb. I think it's running about how long it takes to empty bowl? Bowl not refilling? or not full to begin with. Carb King will know. Some ball left out, accelerator pump, needle, float.........idle circuit.
At least half of the lube is already wiped off the cam and etc. Usually the break-in is after you start the motor and run. I don't think trying to start it is a problem, but when it stays running, then the break-in procedure is on. (whatever you have been told to do by the rebuilder).
You have to to get it started. There is no way to get it started without trying to get it started. Your cam is OK as long as your oil pressure came up during the one minute runs. Limit the starting until something major is discovered or done. Like: Jerry's thing or another carb. Slapping a running loaner carb on there takes 10 minutes. Got any buddies with a carb? Looks like you are pretty isolated back in there. That is why I was suggesting the squirt thing. If you were sure, you would order a new carb, fix the other later and sell it.
If the hose is not genuine fuel hose, it will collapse from suction. That will stop the motor. If the hose sucks itself to the side of the gas can, that will stop the motor.
Logic says: Wouldn't expect a rebuilt carb from J. Carter to be bad.
Logic says: It's something we didn't expect to be bad.
Just trying to think outside the bottle.
Hope Jerry is right. That would be wonderful.
Be careful of voiding any warranty.
Tim, I see you got this, you are doing everything right. As Bugs Bunny would say, you need "west and wewaxation".