The Avengers deserve the right to choose for themselves without a third party telling them they can or cannot go on certain missions. Like Steve mentioned in the film; "We sign this, we surrender our right to choose". Since when do you need permission to save someone? If the government thinks there will be less collateral damage if the Avengers are controlled by the UN, they're wrong in my opinion. It's not going to make much of a difference.
From the start of the avengers to right now, the Avengers and its members keep casualties to an all time minimum by without government supervision. I know there were some casualties but overall, the planet is safe from domination, destruction etc. More importantly, the safest hands are still their own.
Cap is on the right page here. I can see why Tony wants to be supervised but the Avengers are sensible people who won't put people in harm's way when it's not necessary. They shouldn't need permission to save people. Does that sound logical to you? The government can get corrupt and stop the Avengers from doing right. This is ridiculous. No permission required. Yes, they might cause over a million dollars in property damage but which is better may I ask you: a few million dollars in property damage or a world where the human race are dead?
Members of the government will likely try to manipulate the avengers for their own agenda and prevent the avengers form doing their jobs. The avengers know how to do their jobs better than the government. They have saved more people than killed. Steve was right tony was wrong. They should stay private.
The Avengers is made up of a few personnel that are tightly knitted comrades. If left on their own, the organization of superheroes made independent decisions within their narrow perspective, and may be myopic to other considerations. It is not a secret that Steve Rogers, or Captain America is willing to sway the entire organization to protect the Winter Soldier Bucky, who is also emotionally tied to him as someone he knew as a friend for a long time. If decisions made by a group of vigilantes who wield exceptional powers are unregulated and governed by emotions rather than reasoning, we simply cannot trust the Avengers to do the best in a given situation. Thus, for the greater interest, we should have the Avengers in the UN.