Why do gas water heater exhausts have an air gap before the chimney?
The actual amount of combustion air - i.e., air flowing past the burner to provide oxygen and leave with less oxygen but more CO2 and possibly other combustion byproducts - is very little. The gap allows additional air to flow in and the combined air, if the ductwork is designed and installed properly, flows through the ductwork and out of the house.
If you did not have the gap, the amount of air naturally flowing through would often be too little to flow at a decent speed and would instead be more likely to sit in the ductwork. Hard to explain, but it just wouldn't work right.
A few additional points:

CO2 is not the biggest enemy - CO (Carbon Monoxide) is.
Yes, if you have too much CO2 you will have problems (think Apollo 13). But any house, even without any combustion equipment, needs to get makeup air with more oxygen to offset CO2 produced by people. But CO is incredibly dangerous and even low levels can cause a problem. CO is primarily produced by incomplete combustion which can happen easily in any combustion equipment if air flow is too low or there are impurities in the fuel or if a burner is clogged or many other problems. Therefore, you should have CO detectors near any area with combustion equipment.

Temperature of the output is actually a different problem than it used to be.
It used to be that furnaces and hot water heaters were relatively inefficient and the exhaust would be hot and you would want to make sure it all got out so you wouldn't burn your house down. But since it was hot, and hot air rises, that was easy to do.


Now furnaces and hot water heaters are so efficient that there is a different problem - the exhaust air, which you still want to get rid of because of CO2 and other potential problems, is too cold! So it doesn't rise as well on its own. That places a lot more rules on how to vent the exhaust, to the extent that where in the good 'ol days you would replace a furnace or hot water heater and reuse almost all the exhaust ductwork, now there is often a lot of work involved. But I digress.