Purpose of voltage amplifier (VAS) in three stage amplifier design
The second stage obviously provides additional voltage amplification.  However, it has an additional function as well.  The capacitor from collector to base provides the dominant pole compensation that makes the amplifier stable (in general from unity gain to very high closed-loop gain.)

Since this is likely an integrated op-amp, where capacitance take a lot of area, the compensation takes advantage of the Miller effect to multiply the effective capacitance by approximately the gain of the stage.

The Miller Effect

Also, the PNP input stage allows the common mode voltage range to get much closer to the negative rail.  Could be useful for single-supply application.