Why are civilians not allowed to fly demilitarized US fighter jets?
That's because US military is particularly concerned about chopping up its assets before civilians can lay their hands on them. The procedures are covered in US DoD Manual 4160.28 Defense Demilitarization: Procedural Guidance.
The procedures for 'demilitarizing' military aircraft are particularly brutal:

Military Aircraft
(a)  Aircraft (Figure 29 and Figure 30) that are specifically designed for military purposes shall be demilitarized as follows:  at a minimum, cut or break completely through at least one lobe of the fuselage trifurcated horizontal and vertical stabilizer spar attachment fittings, on both the right and the left hand sides of the stabilizer carry-through spar assemblies.  This demarcation of the prescribed DEMIL procedures is to ensure the aircraft is rendered unfit for flight.

(emphasis mine).

Demilitarization Procedure; image from Popular Mechanics

(b)  Helicopters (Figure 31) specifically designed for military purposes shall be demilitarized by crushing, shredding, or smelting the entire airframe and fuselage, ensuring that the transmission mounts and supporting structural beams, engine deck in area of mounts, wing attaching mounts and support beam structure, and fuselage to tail boom attaching mounts and tail rotor gear ox mounts have been destroyed.

So basically, it boils down to the fact that the US military makes sure that the airframes won't fly again. Of course, there are ways around this- you can assemble the aircraft from parts or some aircraft will simply escape the procedure and is avialble:

In a letter to the chair of the House Armed Services Committee, AOPA President Phil Boyer ... stressed that certain aircraft (possibly aircraft not demilitarized) had been carefully maintained over the years and were now serving useful purposes.

but these are quite rare.