One interesting details I've found about Spider-man being on teams (when he's well written) is that on a team of older veterans, he's the trouble maker and the wiseguy.
On a team of rookies and young heroes, he tries to play mentor and voice of reason, whether they want it or not.
I think trying to walk the line is a vital aspect of his character, much like how he tries to balance his time as Spider-man and Peter and often ends up focusing too much on one at the expense of the other.
The idea that he would STAY a member of a team for a long period of time isn't one I can get into. He wants to do his own thing too much.