ACH transfers are the evolution of paper check clearing houses. Transactions are conducted in bulk and do not immediately settle -- the drawer and drawee still retain liability for a period of days or weeks after the transaction date. (I'd suggest looking to the legal definition of a check or draft to understand this better.)
A for-fee wire transfer still goes through an intermediary, but settle immediately and irrevocably. Wire transfers are analogous to handing cash to someone.
In the US, the various Federal Reserve banks are involved because they are the central banks of the the United States.
In the past, bank panics were started or exacerbated when banks would refuse to honor drafts drawn on other banks of questionable stability. Imagine what would happen today if your electric company refused to accept Bank of America or Citibank's check/ACH transactions? Wouldn't you get withdraw every penny you could from BoA?
During the 1907 banking panic, many solvent banks collapsed when the system of bank "subscriptions" (ie. arrangements where small town banks would "subscribe" to large commercial banks for check clearing, etc) broke down. Farmers, small business people and individuals lost everything, all because the larger banks would not (or could not) risk holding drafts/checks from the smaller banks.