The vacuum line is directly hook to the intake usually. Once you shut off the engine the vacuum drops immediately and closes the petcock. If you have a hole you would never start or run.
If it starts easy hot but does not cold 2 issues...
#1 Fuel drains from the carburetor? The engine starts with the remaining fuel after you turn it off. Once the engine starts the vacuum is there and opens the petcock for fuel. You are off and running.
#2 the Auto Choke is a very odd option on these scooters. It is electrical and when cold it chokes the engine to start easy the first time and then after a little time it goes back to normal. You need to check that this part on the carb is working right for the cold start.
Never turn the engine over without a spark plug in place you could ruin the CDI Box.
I hope this helps. It has been a long time since I worked on a scooter but this is what I remembered from the one I did repair.