> Would it be safe to say that CIA (and NSA and other parts of the intelligence community) is the actual "State within the State" (i.e., the so-called "Deep State")?

This shouldn't be done or said because it absolves the state of it's absolute willingness. When the CIA backed Pinochet, the state was absolutely interested in getting rid of Allende, and the same can be said of any other CIA-backed coup.