I'm glad we went for the Director of Football approach especially as in the past we've had managers with a very large remit, the problem with the old 'manager does everything' system is you are assuming that the manager will be there for the long term and with the example of Martinez, he seemed to have quite a large say in how Finch Farm was upgraded with extra bedrooms or whatever it was he felt suited his philosophy. He's gone and the current manager has a completely different way of looking at football which risks overhauling the system influenced by the previous manager.
For me the Director of Football should provide continuity on the footballing side of the club and help Everton establish some sort of consistent footballing principle or philosophy that reflects the development of the club. So when a new manager comes in after the old one is sacked or moves to another club, they have to be able to fit with the Everton ethos and make the most of the existing players at the club whilst also working with the DoF to come up with suitable transfer targets.
I'm not sure the board have this DoF structure fully sussed out as it seems that Walsh is focused mostly on the scouting side and I don't know what other responsibilities he's supposed to have especially regarding the actual footballing side of things. I think Kenwright said something along the lines that he still thought the manager was the most important individual on the footballing side although I don't know if Koeman is directly answerable to the DoF.