The Apache was never designed to handle air targets. It's meant to be a ground pounder, relying on the Air Force to handle the air. Or Patriot, Avenger, etc.
Apache fills the role of a heavy duty VTOL tank buster. I'm not even sure there's hardpoints for sidewinders on it unless you swap out the hellfire mounts for sidewinder mounts. But, ultimately, not much of a need with all the other assets available.
The Cobra has to be able to take down air targets as the Marines, typically, don't have as many assets to rely on. They have to take what they have and make it as universal as possible. I'm no Marine, but I'm pretty sure I hit the nail on the head there.
As far as the game goes... If you want sidewinders just mod them in. I think the Apache outclasses the Cobra in just about every aspect but I can't disagree with the Marine doctrine of making everything you have cover as many roles as possible.
I could live with a pair of sidewinders on an Apache. Just don't touch my eight hellfires.
And as far as missiles go... The hellfire is big and fat. It's probably slow, too, compared to the Sidewinder. I'd imagine it doesn't have the range or mobility that a sidewinder has, either. Even if hellfires could lock onto air I probably wouldn't try to hit with one. Jets are nimble, tanks are not. Different roles, once again.