I don't know the reason for this technique in the F-4 but I have what seems to be a plausible explanation. With a tricycle undercarriage aircraft, when initially rotating on the ground the point of rotation is the mainwheels. As the aircraft unsticks, the point of rotation in pitch becomes the centre of gravity which is forward of the mainwheels. Therefore, the horizontal stabilizer instantly has a longer moment arm resulting in an instantaneously larger nose up pitching moment for a given stabilizer angle. This could result in an unpredictable response in pitch. However, using the F-4 technique, as the nosewheel starts to lift off the runway the pilot is given a cue to smoothly move the stick forward which results in a smooth capture of the desired pitch attitude after unstick. I suspect that it is just that the pitch response characteristics of the F-4 in this flight regime make this a good technique. Also, it is simple and calculation of a rotate speed is not required.
However, there is the potential for the horizontal stabilizer to scrape the runway during rotation. On one of my very early sorties in the F-4 I was flying a totally clean F-4K which thereby had a very aft c.g position. It rotated more rapidly than I had seen on my previous sorties and although I moved the stick forward more quickly the stabilizer tips still scraped the runway. Luckily, the rubbing strips worked and there was no damage. I was not the first to do this so we refitted a ballast Sparrow on the front station!
Full aft stick take-off rolls are also flown on Hunter post-maintenance test flights to check the nosewheel lift-off speed although the pitch attitude is then checked and a normal unstick speed used.