No, how a DoF should work is have a vision for the club and appoint a manager befitting that vision - including in terms of style of play etc.
So having appointed EtH, I’d assume the DoF (& the back room team) would draw up lists of players who fit into the clubs (& ergo the manager!) ethos. ie shouldn’t matter which striker on the list say, 1-5 we sign, they should all have the characteristics of how we want to approach our football. So it’s not just randomly going out there and signing whoever EtH wants.
The crux being, when/if EtH is replaced, we need to ensure we identify a list of managers with the same/similar ethos and thus we wouldn’t need to replace the entire squad again, as we needed to going from Moyes, to LvG, Jose, Ole & EtH.
So unless it doesn’t end up like this and go to appointing Conte after EtH, there’s no way you can accuse them of still following the old blueprint.