Directors like to consider the big picture. They don't like to be bothered with details. Management mayhem ensues. How can it be prevented?

Actually that is a very legit way of problem solving in which seen the whole picture helps seen each and everyone contrinution, for example a actors only see their part as the director is only seen the whole picture.

it should not be prevented, each line manager responsible to see his own section, and each section responsible to see their sectors, so leaving small details for them to give presentation to the director