Look at the schematics for the Foxx Tone Machine, Univox Superfuzz, Honey Fuzz, Ace-Tone Fuzz, and a bunch of others and you'll see many similarities. There is the phase splitter (transistor with equal-value emitter and collector resistors, providing two opposite-phase outputs), the diode pair to ground, and the notch/scoop filter on the output.
The big differences between the Kay and these others lies in the input stage. You will note the absence of any network linking the base of the input transistor and the emitter of the phase splitter (Q1 and Q2, respectively). I don't know a lot about discrete circuits, but I take it that this network has the function of providing more boost for certain portions of the spectrum, and especially boosting the fundamentals that are likely to make up the doubled frequencies. The Kay is a simpler circuit that omits this element, which, in principle ought to make for a thicker fuzz with a less obvious octave up (hence no means for selecting/deselecting it).
C9/C10, R14/R15 make up the traditional midscoop filter in these pedals. The values are fairly similar to those seen on the Superfuzz except that the value of C9 is substantially larger, allowing more mids through to the output. You will note that C9 sort of steps over the pot directly to the output, such that mid/treble content is generally uninfluenced by the setting of the pot. I wouldn't say completely uninfluenced, since you will note that the output of C9 has two direct paths to ground: one through R16 and the other via the pot wiper through the ground leg of P1. Place those two resistances in parallel and that resistance plus the value of C9 forms a kind of crude variable highpass filter.
At the same time, consider that R14/15/16 form a kind of fixed voltage divider, kind of like a 132k pot turned down partway (32k on one side, 100k on the other). But whoops, once again we have this "other" path to ground that places the ground leg of P1 in parallel with R16. We also have another path whereby R14 and C10 form a lowpass filter with a corner frequency of 72hz and P1 simply attenuates that output.
So, P1 seems to play multiple roles that involve attenuating the bass, shifting the highpass corner frequency, and attenuating the overall signal level. As noted, moving the pot around, whether by fingers or foot, will produce equal parts volume and tonal change. Not having used one of these, I have no idea what the effect-vs-bypass volume shift is. Given the complete absence of any post-filter gain recovery stage, such as found on both the Superfuzz and Tone Machine, my sense is that the output will be a bit louder than bypass, but not by much, making the pot more of a tonal change than anything else. This is likely part of why it is situated in a treadle mechanism like a wah rather than a panel-mounted pot. If the volume won't change much then the player won't get themselves into trouble by turning up too loud and risking feedback.
This pot is actually a nifty little control, and entirely capable of being implemented on many designs that use a gain recovery stage. For instance, the Superfuzz has a 100k terminating resistor on the output stage. That could easily be converted into a 100k log pot for volume control, and the volume pot normally situated before the output stage co-opted into functioning like this little-bit-of-this-n-that control found on the Kay. It might actually make for a more interesting tone control than the stock one found on the Superfuzz.
