It's a common conceit of superhero movies that the threat has to be stopped by the hero, and the military doesn't play as large a role as it probably would in the real world. Out-of-universe, it's an understandable narrative device.
That being said, given the role that the Helicarrier played in the first Avengers film, their lack of involvement in the Battle of New York seems absolutely bizarre.
- We know it was very close to the battle, because the nuke-delivery fighter was almost in firing range before Nick Fury could even radio Tony Stark to warn him.
- We know that they had planes and quinjets available (i.e. they didn't all fall off during the Helicarrier's almost-crash in the Second Act), because we can clearly see several on-deck when Nick Fury tries to stop one from launching. That video also shows relatively little deck activity (meaning they weren't prepping to launch any time soon).
- We know that they were capable of launching from both runways, because one did from each, and it didn't seem to take any special repairs or effort to make that happen. We also know it didn't take long, because one was in the air before Fury could stop it.
- We know that the SHIELD agents aboard the Helicarrier were watching the battle unfold, and weren't tied up with other efforts aside from general repairs.
What explanation could there possibly be for not sending in those fighters? Even if the Chitauri sleds were too fast and maneuverable (although it seems worth a try), the Leviathans would have been easy targets for missiles and cannons.
We never even see them discussing sending help, that I can recall. Do we know why Nick Fury didn't offer any air support to the Avengers?