Just watching Midway, and in the scene of Raymond Spruance taking command of the Midway Task Force, I remembered an old question that nagged me. Apparently, the US Navy equipped task forces and fleets with whole divisions of cruisers and destroyers as escorts. These divisions had their own commanders and staff carried on some division flagship.
Given that a division has typically consisted of 4 ships each, that seems like an awful lot of staff per task force. At the battle of Midway, that would imply 2 cruiser divisions and 4 destroyer division staffs. So why did the navy keep this organization in war time (or did it do so only in exceptional cases)? What was the purpose of e.g. a cruiser division staff attached to a carrier task force (which would probably never send her cruisers on a detached mission)?
edit: To add some more precision to my question: I understand the general concept of a chain-of-command that splits larger forces (armies, fleets) downwards into smaller pieces (corps, divisions, regiments/brigades, battalions, companies, platoons). However, in the army, if one of these formations has its own staff (e.g. a division or regiment), that staff has a clear operational role, e.g. a particular area of the front to defend. A task force is inherently a combination of different ships specialized in different roles (e.g. a destroyer squadron might be attached for air defence or ASW purposes). So when a cruiser division is attached, would the division's staff be handed over the task to deal with that division's role?
E.g. "Adm. Spruance, take your division and protect us from enemy cruisers! Adm. Foo, your destroyer division handles air defense for us and the cruisers! Adm. Bar your destroyers catch all submarines that might interfere with us, Adm. Spruance or Adm. Bar! Oh Admiral Baz, I forgot you command all the destroyers since coincidentally, they form a squadron. Sorry, but I have no task for you."?
Sorry for the ludicrous example, but I really have a hard time understanding how such a command structure would work during a real operation. Since different roles need to supplement each other, every staff would have to reconcile its plans with the all the others in the task force. That seems to imply a lot of friction. Why not task one (sufficiently equipped) staff aboard the flagship with all that?