In the old days there was only one team of Avengers. As with most teams, the roster was constantly changing. With so many different members over the years and the need to fight evil on a bigger scale, the West Coast Avengers was formed. That gave two teams. Eventually an attempt was made to establish a team in the middle of the United States and the Great Lakes Avengers was created...but never had an official sanction.
With the passing of the Super Human Registration Act, a civil war came about within the super-powered community. The heroes were split and former teammates found themselves fighting against one another. This lead to Captain America's Secret Avengers. They believed in the human rights and privacy of heroes. They stood against the government and the decision that all heroes should register and reveal their true identities.
Once matters were settled to a degree, we still had two separate teams, the Mighty Avengers and the New Avengers. With different goals, it made sense that all these former Avengers would continue fighting in what they believed in, even if on separate teams following different ideals. With Captain America in charge of all the heroes, is there still a need for the separate teams? The Avengers is the main team. The New Avengers is still sanctioned by the government but supposedly follow their own lead. The Secret Avengers was meant for black ops missions but it didn't feel as if these missions really needed to be a secret. Of course there's also the Young Avengers operating on their own..
With so many different Avengers under Captain America's command, would it be best to just have one team rather than three separate ones?== TEASER ==
The dangers the Avengers face can't always be manageable for each team. They have a wide variety of villains ranging from regular street villains with some sort of gimmick to galactic and cosmic powered threats. An existing team of Avengers or New Avengers might not be the best prepared team, regardless of the amount of experience each member brings. The Avengers don't go out on patrol so having a set roster isn't necessary.
What would make more sense is for Captain America (or whoever is in command) to assess the situation as it occurs. Once the threat has been determined, the appropriate members could be gathered and dispatched to deal with the enemies. There could be a loose established roster of heroes with a variety of powers (strength, speed, intelligence, etc) that could be sent on routine missions or ones where the full extent is not known. Once the details are known, the Avenger(s) best suited should be immediately sent out.
Having a large number of Avengers available would require a facility that could house them all. Some Avengers living in the Avengers Mansion while others were at the Avengers Tower (before it was destroyed) doesn't make sense. There's no reason they can't all be in the same place (provided they have an incredible security system that won't allow the entire team to be wiped out). Then if Doc Ock or a former Herald of Galactus shows up, whoever is needed could be sent out.
There is an argument to be made in having separate teams with actual rosters. An established team relies on teamwork. Once they know each other and their powers, different strategies can be established. Working together is the whole purpose of banding together. Most of the Avengers have faced supervillains on their own. Knowing how to work as a team will make capturing the villain easier and faster while also ensuring the safety of innocent bystanders.
Still, having separate teams can be limiting. Why should a team only be allowed a set number of members when there are so many other Avengers that could go with them? The more the merrier. There's also the fact that some Avengers appear to be on multiple teams. Spider-Man and Hawkeye, for example, jumped back and forth between the Avengers and the New Avengers. They should just go with the team that needs them.
There is a wide range of experience among the Avengers. Many have worked together in some capacity, whether on the Avengers or in simple team-ups. Many of them will know how to work with each other. They will know each other's strengths and weaknesses. Being able to rely on each other could make a huge difference in the middle of a battle but because they are all professionals, the Avengers won't allow just anyone to become a member, they should be fully aware of each other's abilities.
As long as Captain America can arrange for random training sessions with different groups of members, the idea of having one super-team would be ideal. The bad guys don't schedule attack times. If one team is fighting villain and another strikes elsewhere, expert coordination would be needed to make sure the best suited Avengers are dispatched. They are all Avengers after all. Why do they feel the need to keep themselves separated in separate teams would the world would benefit from one team dispatching members when and where they are needed?