Why do envelope extraction before pitch detection?
My guess is that the Hilbert Transform is a pre-processing step which attempts to make the actual pitch detection stage (AMDF, in this case) more robust.

The basic reasoning is this: pitch detection of speech is made more difficult by the time-varying spectral characteristics of the signal.  On the other hand, pitch detection of an impulse train is easy: just count the number of samples between each pulse (or look for the first peak of the auto-correlation function/the first dip of the AMDF etc). Pre-processing the speech signal to make it more pulse-like therefore improves the results.

In your diagrams, the Hilbert transform does make the speech pulses more impulsive, though I think that other pre-processing steps (e.g. inverse-filtering by the estimated spectral envelope) is likely to yield better results.