I think we have to choose first if there is a team existing at the start of the story (and thus with a name and a leader) or if the team is formed at some point of the story.
If it's the latter the name and the election of the leader would become matter of the story.
As for the scheme for the leading role, I think first we should see what kind of team is: if it's an underground group of outlowed vigilantes a rotating leading seat could be good, while if it is a global spanning peacekeeping force I guess the election of the leader will be a political matter, with the bigger countries of the world trying to impose their will.
MY scheme for starting the story would be: we create a sort of threat, that normal police forces or military can't solve by themselves, that cause the characters to meet, let the story play, and at the end we decide: if it was a sort of one shot, there is not even the need to find a name for the team, while if we want to carry the group of characters for other stories, we find a status and a name for the team.
