Unphased by EQ

I have had a pre-amp with an 8 channel DSP for years - with individual volume for each channel and a master volume too. The volume is digitally controlled but analog - so no loss in bits.
Why do you get a change in phase because of a change in gain?

You won't electrically on a single channel for sure.
And I'm not an expert on IIR xovers by any means, but the electrical measurements I've made of summing the sides of simple BW and LR IIR xovers, do not show any phase change either.

But the acoustic output of summed speaker drivers do.
The acoustic xover frequency shifts along with generating a new combined phase curve. (and mag ripples). Damn drivers' have curves of their own lol
 
What interests me more is, if you start from the premise that a small amount of tonal shaping enhances the listening experience when you have a wide range of music of differing tonal balance, the what is the optimal set of controls to have. Too few and you have a blunt instrument. Too many and you spend more time twiddling knobs than listening. Maybe a 21st century equivalent of bass-mid-treble is the right way to go?

Amen!
I'm always looking for the simplest method to get adjustable and repeatable good sound. The simpler the better, till it doesn't do enough.

And your points are well taken re the similarity with the gain per section method, and shelving filters.
Every now and then, I get a track that my method can't quite make right. For example, I need overall bottom end gain:
sub gain works fine, but on the Mid gain i need some boost at it's low end, but can't use any on it's upper end.
Then a normal shelving filter that spans both sub and mid, allowing a select freq, works better.