I don't know Barnes' personality well enough to say whether or not he would make Harden-type leap or if he would naturally take a backseat like Jeff Green. This is not to say that I don't think Jeff Green is an effective player - I always liked him - but Ainge clearly expected more out of him after making the trade. The heart surgery may have made him embrace life rather than obsess with bball.
I do know that Barnes has done an extremely good job settling into his role on a championship team. If he were constantly looking for his own, he would be on the bench a lot more often.
If we ended up signing him, I could definitely get behind it. Signing Horford or trading for Love seem like worse moves than getting a young, efficient, versatile Barnes for 'free'.