I personally think there need to be a good partnership but also some checks and balances between the two positions. Saints fans blame Payton for the cap and the roster, but the traditional role has been for GMs to manage those things and the coaches to coach the rosters they are given. Now if Payton (or DA more recently) did demand certain things that were risky and put the teams ability to sign their own free agents or saddle the team with a player they no longer want, then the GM should be the voice of reason that prevents that in some cases.
That is why I'm not totally against getting them as a pair with a common vision but different responsibilities. What I don't like is the coach making the demands, the GM giving them what they want and washing their hands of the responsibility after the coach has been fired.