If brains are 1000x faster, they type the emails 1000x faster as well.

Why do you think, exactly, that brains are going to correctly integrate into some single super mind at all? Things like memory retrieval, short term memory, etc etc. have specific structures, those structures, they do not extend across your network.

So you got your hierarchical structure, so you ask the top head to mentally multiply two 50 digit numbers, why do you think that the whole thing will even be able to recite the numbers back at you, let alone perform any calculations?

Note that rather than connecting the cortices, you could just provide each brain with fairly normal computer with which to search the works of other brains and otherwise collaborate, the way mankind already does.