I'm actually quite warm, in theory anyway, to this kind of flatter structure. At the moment it seems to me that the DoF can almost be a distraction in accountability terms to individual "heads of" and maybe you are better separating that out into specialised and defined roles rather than having "heads of" in title but not in practise.
For example, Dan Purdy is our Head Of Recruitment. But when recruitment is discussed it's almost entirely the DoF being held responsible. If the actual Head Of gets neither the praise for the good nor the blame for the bad does that encourage performance? Likewise with the Head Of Academy etc.
If the idea is to get the best possible people for each department head, you shouldn't need an extra layer of management between them and board / CEO, that just muddies the waters. Have each Head Of actually carry the can for their individual portfolio and accountability will lead to performance.