For me, the Avengers are >< that far from being a government agency, completely unlike any other Marvel team. Books/teams like the Defenders were created to contrast them. The members should, by default, be "out of the closet" as superheroes. That hasn't always been the case, but the team has evolved that way. Steve Rogers and Tony Stark are no longer secret identities. The Avengers individual members should not ever be characters who are wanted for questioning.
I can go with characters like Luke Cage and Iron Fist, Ms Marvel, Thing (which feels weird, but he qualifies for membership of a S.H.I.E.L.D. sanctioned organization.) Spidey and Daredevil aren't a good fit because their lives keep them in the shadows. I could go with Wolverine as being a former Canadian government agent, he's a loner/misfit/outsider, yes, and that could be played up in the stories, but he basicly qualifies for government clearance.