Because it doesn't have a one man army like Rambo inside. It is ridiculous, a heli should be able to take a city. Air cavalry is used to land troops and create a perimeter.
The helicopter in the game is more representative of an apache attack helicopter, that's why it has a bonus against armor. Attack helicopters don't land troops, personal transport helicopters do.
The closest thing to air cavalry in the game is a paratrooper. And both paratroopers and air cavalry are basically foot soilders once on the ground, the only difference is how they are inserted into the battlefield.
The same argument for helicopters could be made for ranged units. If the city health is down to zero and there is no unit in the city, why can't an archer or artillery just walk right in and take it? I find this much more irritating than not being able to take cities with helicopters.