The lungs aren't filling up with that much blood in the short amount of time it takes the deer to pass out.

What is causing the deer to stop and fall is lack of oxygen to the brain. The heart is what supplies that oxygen rich blood. As you said, regardless of a heart or double lung shot, the lungs are taken out in either so I think the deer sucking blood into the lungs it is a moot point. It takes a couple of minutes to drown. Only seconds for the heart to pump out enough blood to starve the brain of oxygen.