It looks like this gain control functions by dumping negative feedback (NFB) to ground. When you turn the gain up, there is less NFB because there is less resistance between it and ground. This is the opposite of a normal volume control, so I don't think a bright cap would work in this circuit.
When the gain control is at 0%, there is more NFB getting past the gain control, and that NFB is killing proportionally more lower frequencies (below 720Hz).
When it's at 100%, there is less NFB getting past the gain control, and the NFB killing proportionally less lower frequencies (below 62Hz), so it *ought* to sound "fatter" when it's turned up.
I don't understand why it would sound like it were rolling off highs. Maybe because less gain = less distortion = fewer harmonics/overtones that would fill out the high end?
Looks like you are going to need someone more experienced than me to answer this.

I say to try two things: 1) try the bright cap like you suggested and just see what happens. It's worth experiementing. And 2) try removing that gain control (disconnecting the .047 cap from the op amp) and replacing the 1M resistor with a pot. That is a more standard gain control. You might or might not like it better in this circuit.