To me, the stupidest thing United have done since SAF's departure was replacing Moyes with Van Gaal and Van Gaal with Mourinho. That makes no sense: it's three different styles in just a few years, making long-term player recruitment impossible.
So to me, when people speak of United DNA, that's what the DOF should define: on a high level, what should the club play like? Possession football? High-pressure transition-focused football? Counter-attacking from a defensive setup? Something else? Do we want to be a development club or buy to win? And there are probably some other high-level things to determine. Some of this is also budget-dependent and would need input from the CEO.
Once that's figured out, the DOF oversees the hiring of a coach that fits that general vision. Obviously, the coach would fill in the details in their own way; a DOF is not the tactical mastermind. Also, the vision can develop, influenced by the coach and other factors. But this guarantees that the club can direct all of its resources in the same direction: youth development, player scouting, and indeed coach scouting - replacing a coach isn't as big a deal if you roughly know what you want and always keep a shortlist of potentially interesting coaches.
From that general setup, the rest can be figured out as required. For example, some coaches might want to be closely involved in player recruitment, some might not; and the DOF might have their own preferences for the decision-making process around recruitment. They might also want to keep a coach out of the process as much as possible to ensure they can spend maximum time working with the first team and preparing for matches. (In that sense, the model also allows coaches to be more focused on the football.) And so on. But the main thing is guaranteeing direction - and then this Moyes-Van Gaal-Mourinho nonsense won't happen.
At least, that's my takeaway.