I broadly agree with Nate on this. Someone not a DoF has at least gotten through to Levy that Fabio's ideas about managers were rubbish and has been finding the right kinds of people. So, all the structure changes in the background this year seem to have started to produce results, even if it's not fully operational yet.
Get in the new manager, work with him to evaluate the current squad, and add in the DoF when you can, and everyone is broadly on the same page already. Everyone knows the squad is a train wreck, and everyone agrees we need to get rid of a lot of people. And with the the managerial candidates all wanting to press and play possession-heavy football, you will have a pretty similar list of guys to keep and transfers to make, even if the order of preferred players might vary slightly.
And Daniel Levy is out of the loop on all of that. Even if the budget isn't perfect, it's the kind of situation that will work.