In the first movie they banded together to save people and stop the planet's invasion. They kept damage to a minimum but they did not notice people stealing alien technology.
In the age of Ultron like Adrian said Iron man created a dangerous AI without talking to anyone or even taking precautions. He almost singlehandedly created an AI apocalypse. And then I think Hulk lost trust when he found out about the Hulk Buster armor because he quit the team afterward.
They dropped a city on a city and people were killed because of Tony's mess.
Scarlet witch does a small mistake but that is enough to change the opinion on them.
Tony wants all of them to become patrolled and kept in check by the governments.
And you can't do that because certain dictatorships and regimes would prevent the avengers from going to certain countries.
Plus the identity of the heroes would be made public and they have a lot of enemies.
Iron man is trying to punish himself for his sins but he is the one who breaks up the team when he wants to kill a person for killing his mom and dad. He causes half the world to die and he gets to have a family while Peter Parker was snapped away. That is probably why he used the stones because he causes more trouble than good.
And why didn't strange have a better plan than tony snapping his fingers
Exactly like Cap said, If the Avengers were apart of the government then when they need to go places which could be to help people, They may not be able to do that under their authority. What if they need to go places and they won't let them? I am fully aware that they have caused some innocents to die but they've done plenty good like saving the planet from the Chitari. And don't forget, Tony was the one who made Ultron which caused this debate to happen. The only other reason this event could've taken place is also because of the incident with Scarlet Witch. So instead of placing the worlds strongest people as puppets for the government, They could have a system to train people with special powers so that incidents like that could never have taken place. If anything, Tony should be placed in check and not everyone else due to him, Again, Creating Ultron. And the government could also just ask S. H. E. I. L. D to keep watch on them and if anything like that happens again, Then they can face the consequences.