Director of football makes sense. It allows the manager to focus on the team, the training, and the tactics, while the director of football can manage scouting and transfer dealings.
A good DoF will work very closely with the manager to ensure he's focusing on players of the type that the manager feels are needed for the squad. One man shouldn't run literally every aspect of a club - the job is too much for the vast, vast majority to handle.
Is it just England that has a fascination with the do-everything manager? A lot of the 'great' manager in the history of English football had that sort of relationship, just named something else - Taylor and Clough, Shankly and his boot room are two obvious examples.