I must say, it's been fun reading everyone's ideas and responses on the subject. I'm thrilled that we can have a rational argument about something totally fictional that simultaneously has deep rhetorical repercussions. If only such courteous decorum could penetrate real-life politics as well.
I'm with Cap. Many of my reasons have already been stated by others, but here are a some that I think hold some serious clout.
To begin with, the ideas and rhetoric behind registration, though seemingly well-intentioned (based on the ideals of safety and security) sound a lot like the Hydra hogwash from Captain America: The Winter Soldier. The repeated argument is that people need to feel secure and safe and registration will provide that. Let us remember that Benjamin Franklin said that ”People willing to trade their freedom for temporary security deserve neither and will lose both.” Freedom is the first and foremost of American ideals, and safety, should we have it, ought to come as a consequence of freedom. The one component of Team Iron Man's argument that I can almost agree with is the concept of holding super heroes to the same standards as law enforcement where interrogation and imprisonment are concerned, but for the sake of the rights of those involved, paranoid security. However, we need remember that super heroes take on the baddies that regular law enforcement can't handle and usually in a time frame that doesn't allow for due process, 'cause, ya know, the world might blow up or something.
Which brings me to point number two.
Remember that swell guy, Thanos? Or that other bloke, Loki? Those chummy pals? Hydra isn't the only thing we have to worry about around here. There are a lot more and far worse bad guys in the universe, and sooner or later they will turn their cross hairs on planet earth. That's why we need super heroes acting independently. In the Marvel Universe, we have to acknowledge and deal with powers we can't begin to understand nor control, yet for some reason the government seems to think it's all about busting up gangs and drug cartels or something. We need to consider the threat of global destruction that involves things like Chitauri space whales and infinity stones. How can we expect standard government training that will teach super heroes how not to throw semis at each other to prepare them (or us) against such a threat? They need the ability to adapt and improvise in any given situation at any time without fearing retribution from the society whose collective neck they just saved.
Fact is, there is no perfect solution where people like Bruce Banner, Captain America, Scarlet Witch, and friends are involved. We can't hope for, nor should we seek a perfect solution. What we have is people with extraordinary abilities that put their lives on the line to save the rest of us. They all have crazy, complicated lives and a lot of baggage we can't understand, so let's just thank them for saving our planet and not blame them for damage inflicted by the guys they're trying to stop.