Has the "Foote takeoff" technique really existed, and how did it work?
The 1952 takeoff accident you read about was piloted by Captain Harry Foote. The technique got its name after the accident, which Captain Foote was blamed for (Comet! The World's First Jet Airliner, page 125).

Pushing the nose down is not standard though, rather a corrective measure to over-rotation past 6°. The article quoted below covers the proper technique and the 1952 accident:


  The B.O.A.C. Training Manual recommends the following take-off technique:
  
  
    At 80 kts. the nose should be lifted until the rumble of the nose wheel ceases. Care should be taken not to overdo this and adopt an exaggerated tail-down attitude with a consequent poor acceleration.
  
  
  (...)
  
  Take-off tests by the manufacturers have shown that a constant 6° incidence of fuselage during the ground run gives good results for distance run and for climb-away behaviour. They have also shown that an increase of incidence to 9° results in a partially stalled wing inducing high drag which appreciably affects the aircraft’s acceleration, and that the symptoms are noticeable to the pilot as a low frequency buffet. The aircraft recovers from its semistalled position if the nose is pushed well down. [emphasis added]
  
  (...)
  
  It was the opinion of the investigators that the accident was due to an error of judgment by the captain in not appreciating the excessive nose-up attitude of the aircraft during the take-off.
  
  (flightsafetyaustralia.com)


As @RAC said, VR didn't exist. Judging by the historic FAA regulations, VR and the takeoff speeds in general for Part 25 came about in the mid-60s.