Thats correct, and that is the source of oxygen for the gas flame. So then, why did I think this would work at all?Yea, it will draw from the path of least resistence, so if his box has a hole under the burner for it to draw air, and serve as a bottom vent, That's where the air is going to enter.
There are two intake vents, one on each side at the wood chip level of the smoker. I reasoned these vents were there to supply fresh oxygen to allow the wood to burn. When I closed the vents the temperature in the smoker, without wood on the chip tray, lowered by somewhere around 8 degrees. i was only able to close them partially. It wasn't clear how much flow I would get from the vents, but it is some. i have closed off these vents, attempting to replace them with the mailbox.
i didn't know if I could change the thermodynamics of the box sufficiently to get enough pull to move the smoke from the mailbox to the smoker. As a last attempt, in the original plan I would add a convection oven fan to either push or pull the smoke into the smoker.