Interesting, someone else just had this same question on Facebook.
You do not have to wait for other players. Each success at the scene, by anyone, turns over another shot marker. When all three are turned over (or two, or one, depending on the set) the scene is finished and you are free to go.
In the meantime you are still working the SAME ROLE. You are not finished with the role until the scene is complete.
The rules actually DO say "you can't move off a role once you are on it." Your disconnect comes from assuming that you are done with the role when the scene is only partially complete.
Hope that helps,
Doh! I was specifically looking when we played it to see whether there was a rule that said you had to stay in a role and didn' t find it. After reading your reply I re-read them and it's there in two places.