Over the years all the evidence has pretty much confirmed that Wenger solely dictates any sort of policy regarding playing staff. I seriously doubt anybody else at the club can check that control he has, he'd resign on principle if control over matters such as contracts, transfers and outgoings was ever comprised.
Alot of the club's current policy is a reflection of Wenger's own personal ideals. Lets look at the his "socialist" wage structure, which is largely motivated by his humanist ideals in practice in football. That isn't an Arsenal policy at all, it a driven only by Wenger but he's allowed to do it because of his job security and his position over football matter at the club.
The next guy after Wenger is hardly going to get a message from the board instructing him to continue maintaining Wenger's ideals in practice or else. It depends on how we bring in but if we go for someone like Eddie Howe I suspect he'll delegate alot of that responsibility away but if we get a manager with an actual personality and experience they'd want full control over players matters too.
The dynamic of managing Arsenal will change on so many levels after Wenger leaves, I think people underestimate how drastic the change will be.