Liability, Rules, Politics, & Prestige
The only civil F-4 on the circuit is the Collings airplane. The others are part of the QF-4 program at Tyndall, and called the Heritage Flight.
I can't cite a CFR number, but apparently, there is a law (dating from about the late 1950s/early 1960s) that states no nuclear-capable jet aircraft can be sold/transferred to a private party. In part, that is why it actually took "an Act of Congress" for Collings to get permission to fly their F-4 and A-4.
A 'famous flying person' used his political connections to assist in getting those bills passed. But, when he fell out of favor with the Foundation (for reasons I would rather not publish in public), he tried to use his connections to stop the Foundation. He almost succeeded. Could he still be holding a grudge and be lobbying against them? Hard to say.
Additionally, I think you will find the Air Force and Navy are reluctant to release their former jets as: 1). They fear some sort of liability repercussions if there is a crash, and 2). They want to be seen as the sole-operators of prestigous aircraft. (It's hard to justify to Congress you need hundreds of millions of dollars to keep the Blues and Thunderbirds going if people are willing to sweat at an airshow to see a private F-4 or F-105).
The F-104s on the Starfighter demo team came from Italy (and maybe one or two from Denmark or Jordan; there's one registered in Phoenix). The Sea Harrier came from the UK. Foreign-owned jets were a loop-hole in the acquisition process, so some years back (not sure how many, but let's say post 9-11), a new law was passed, making it VERY difficult to import foreign military aircraft, whether of US or foreign manufacture.
You'll see it on Trade-a-Plane, saying, "...pre-Moratorium aircraft." I admit I don't understand the details nor implications.