Only is baseball do they go in "as" a team member and that is only because on the HOF plaque they have to choose what hat they will wear. Greene went is as a linebacker. Now, he said he's more part of Steelers, but that does not mean it has anything to do with the induction process
From HOF
Is a New Hall of Fame Member Enshrined as a Member of a Team?
Obviously, teams take great pride in the accomplishments of individuals who have been a part of their organization. Often individual teams and even the Hall of Fame will list enshrinees according to the team or teams on which they spent a significant period of time. An enshrinee, however, is not asked to “declare,” nor does the Hall of Fame “choose” a team under which a new member is enshrined. When elected to the Pro Football Hall of Fame, an individual is recognized for his accomplishments as a player, coach, or contributor.