Either the militaries bullets wouldn't work (while Hawkeye's arrows and Widow's bullets, do) making them pathetic or the military would take care of the problem because they have a lot of bullets. There is no convenient, easy fit for the military. The police do mention the National Guard taking an hour to show up right before Cap tells them to keep the streets clear.
If North Korea invaded... America? HAHAHAHAHAHAHAHA! The United States being invaded by anyone is as fictional as super heroes. You don't nuke Seattle, you nuke North Korea, that's the difference between an invasion by a country and an alien force. I expect the Security Council would have shoved the nuke up the Chitauri's ass if they had thought it were possible. Iron Man made it possible.
No, the nuke would not be the first response, the Avengers would
. The military would not add to the tension, they would steal it. What added to the climax was one of our heroes "laying down on the wire" and sacrificing himself to intercept and guide the nuke through the wormhole to destroy the mothership and end the Chitauri threat. How does the military make that ending better?