You are 100% correct. But as stated there is plenty enough fuel in the carb for the initial start to get everything going.
I have a bad memory... I am trying to remember how I checked the auto choke... I think I removed it and hooked jumpers to it from a 12v source. I think I also checked for voltage when starting to see if the auto choke was being given the signal.
If you still think it is the petcock... plug the port where the hose goes into the manifold. Pull a suck on the tube and try to start. If you still have the same issue check the choke. Or you can open the drain for the float bowl, the bike I repaired had this... let it drain till it stops... with the drain still open pull a suck on the vacuum hose to see if you get the carb to drain again. I am sure it will. Then drain the carb again... hook the hose up as it should be for running... hit the started and see how long it takes to get the fuel to drain. It will take a little to drain as there is a lot of distance for it to travel.. So hit the starter for a bit and wait for the fuel.
I hope this helps to trouble shoot it.