Doesn't matter if they bleed out, you collapse the lungs, or they suffocate. The actual cause of death is lack of oxygen to the brain.
There isn't a lot of residual oxygen in the blood. Put a piece of duct tape over your nose and mouth so you can't breathe and then take off running as fast and hard as you can, you won't make it much past a few seconds.
It is really splitting hairs saying they died from blood loss vs collapsed lungs as you have both happening at the same time. The only way to get the heart and no lungs is to get a pass through at the cardiac notches. Rib spaces 3-5 on the left and 3-4 on the right. Even if you do that you still open the chest cavity and the lungs collapse and can't function. You are better off shooting right over the top of the heart. You will put holes in the chest, both lungs and the great vessels as they exit the heart.
For an animal to actually bleed to death quickly you need to hit the heart or one of the great vessels, coming into or going out of the heart. Even then it take more than seconds for them to bleed to death. A mature bull elk will have between 7-9 gallons of blood, about 10% of their body weight. They need to lose 50-60% of that to cause death. It takes more than a couple of seconds to lose that type of volume. Put a 5 gallon bucket in your sink, turn it on full blast and see how long it takes to fill the bucket. Way more than a few seconds.
Pneumothorax doesn't take long to develop. It is immediate if you open a hole into the chest. You open both sides they can't breathe. At the same time you are cutting vessels so they are bleeding while they can't breathe.
Actual suffocation would be rare. For them to physically suffocate the blood has to get out of the vascular space and into the airways and plug them. Would be pretty rare fill enough of the airways with blood to suffocate the animal.