diagnosing darlington transistor problems
I thinks we can not just connect the emitter of the first stage to the base of the second stage without connecting a resistor from the emitter of the first stage to the emitter of the second stage. With this resistor connection, there is no leakage problem due to the first stage.
There is a complex poles introduced by the darlington pairs, that may make compensation slightly more complex if you are designing an opamp.
Headroom of 2vbe is the by-products of the darlington pair, we can not do anythng on this, just to have enough Vcc.