Greg,
You are right. As you turn down the applied voltage to the rails, the three stages of the amp react differently.
The output stage is hardly affected as it is an emitter follower and the supply is connected to the collectors, which do not require a fixed voltage.
The voltage amplifier starts to lose drive, starving the output stage, and the compensation starts to load up the amp, pulling back sonics.
The Long Tailed Pair (sometimes called the diff pair) is the big one. It distinctly dislikes this process. Even a small reduction of stage current due to lower voltage will unbalance the diff amp, creating a lot more H2 and H3 and seriously discombobulating the offset. This is why you must alter R3 to restore the stage current (typically 1mA) so the LTP is again balanced and the offset is within 30mV. At about half voltage the stage current is now so low that one of the diff pair transistors, the feedback side, T2, turns off completely. When this happens, the feedback network no longer functions, and the amp degenerates into open loop, giving gross distortion. Not a nice listen at all!!
Cheers,
Hugh