How does a cascode or beta-enhanced voltage amplification stage work?
The enhanced VAS does offer high input impedance but its main advantages lie elsewhere. In a standard three stage amplifier the Beta-Enhanced VAS increases the OLG ceiling to something like 115-120dB making more available GNFB for the same closed loop gain, resulting in much lower THD and Zout.

Its main problem is awful clipping behavior. But that can be solved easily with a 220 ohm resistor in series with a clamp diode (normal signal diode like 1N4148) connected from the first transistor's base to the second transistor's collector, giving soft-clipping as well as guaranteed latch free overload characteristics. With that added, the enhanced VAS is a clear winner in all the six configs.

Considering the practical application of the configuration, in your simulation of the enhanced VAS there are a few apparent problems:
a)There should be no capacitor in parallel with the emitter resistor of the second transistor.
b)The input needs to be in the form of a current (from a collector-output node e.g. the long-tailed pair), referenced to the negative rail.
c)The first transistor's collector should be connected to ground, not the positive rail.