Enough bullets and you'll hit something vital. I'm pretty sure 120mm armor piercing rounds from a tank will probably be effective against any armor the Chitauri had. The military has a lot of toys.
The Chitauri infantry weren't that well armored or durable. They were the definition of canon fodder but a dude with a shield who does gymnastics, a big angry green monster that punches hard, and a guy with a hammer that can shoot lightning from his ass can only do so much against overwhelming numbers.
The flying chariots would have been a problem for our jets but a couple helicopter gunships would have been pretty effective until they were, in turn, shot down. The Quinjet canon was very effective in the short time it was deployed. More of those would have been helpful.
The only real problem the Chitauri gave the Avengers was that they had overwhelming numbers. Yes, 1 on 1, they are better than a typical soldier, but we have more soldiers than they do (the Chitauri hadn't spread out past a couple blocks by the end of the film). Without the WSC deploying the nuke, the Avengers would have lost, that much was clear. The military would have evened the odds a bit. Whether they would have ultimately defeated the invasion using conventional weaponry is another question.
New York was being invaded. The WSC deployed a nuke that would have killed millions. Oh no, a soldier is gonna die attempting to fight off an invading force from another planet
If there is one thing I have no problem soldiers dying for it's fighting off an invasion of my country. America, **** yeah!
Going back to the original issue, I think by including the military the film would have ended up being the Avengers and the military rather than just the Avengers. The military would have either been stomped, looking weak and dumb (like the civilians and police who already had that role) or they would have held their own and stolen some of the Avengers spotlight. By not including the military, the film ended up being the Avengers v. the Chitauri which looks better on film.
Ultimately I'm not sure why this is an issue. Someone had to have raised this question at some point because there is a line between the police officers that addresses why the military isn't on scene. Mr. Whedon thought about it and dismissed the idea of including the military. If there was no mention of the military in the film, then you have a point. Because there is an acknowledgement of the military, this discussion is pointless.