The Deep Establishment.
I like it! The phrase, that is.
To my mind, The Deep Establishment is magnitudes better than The Deep State.
The latter implies something I stopped believing years ago: that the state is the locus of power in the contemporary world order (or disorder, if you prefer). The state being government, of course (what else could it be?). Government is just political classes — elected officials and appointees — and divisions of administrative underlings doing their bidding. While most at the top may be scoundrels, some (many?) of the underlings are decent people, even if misled. Most will never rise in the governmental system because of their fundamental decency. That alone is a commentary on our times.