Mike, Thanks for looking it over!
A couple of points of discussion--
The circle of fifths is NOT "simple and elegant"-- in fact, it's anything but, as soon as you step outside the confines of the artificially-constructed world of equal temperament! Its apparent logic was back-engineered by Europeans; it's a square peg that's been forced into the round hole that exists naturally in physics.
An equal-tempered "perfect" fifth is narrow by 2 cents compared to a true 3:2 perfect fifth. 2 cents isn't very much, and the ear tolerates it well. But if you were to stack twelve "true" fifths on top of one another... C, G, D, A, E, B, F#, etc. etc. on and on down the line... by the time you got back to "C," it would be 24 cents sharp from the C you started out with (or 23.46 cents, to be precise). This discrepancy is known as the "Pythagorean comma."
Obviously this won't do... and physicists, composers, and instrument-makers wrestled with this for centuries before coming up with the highly inelegant solution of just making every fifth narrow by 2 cents to compensate. Fourths and fifths still play acceptably in-tune; thirds and sixths are quite sour in equal temperament (we've just grown accustomed to tolerating it).
But the thing is... and this is the important part... the twelve-tone scale has never been "arrived" at through the circle of fifths. The temperament came much later. The fact that you arrive at something close to an "E" by stacking true Pythagorean fifths above C is nothing more than coincidence-- the "E-like object" you get from stacking true fifths is VERY different from the E you arrive at when taking the fifth partial of C-- and the third of a major triad was never based on the Pythagorean E.
To put it another way: the fifth partial is so strong that anything out of tune with it will cause beating against the harmonic. You can hear this on a piano... many tuners actually tune in part by counting beats against a timing reference. This is a good time to emphasize that arriving at these pitches is not merely theoretical-- it's physical, and the physics causes audible artifacts in the form of beats against the harmonic. That's how we know "where" each pitch comes from in relation to tonic.
The fifth partial of C gives you an "E" which is 14 cents narrower than an equal-tempered "E." The "Pythagorean" E you arrive at from stacking fifths a la "circle of fifths" (the third partial of A above D above G above C) is 8 cents wider than the equal-tempered E... a full 22 cents sharp of the C natural's fifth partial. This will sound very, very sour indeed, with audible and fast beating against the C's fifth harmonic.
As a side note, we should acknowledge that there are other tuning commas as well-- for instance, if I were to stack three fifth partials on top of one another (C to E to Ab back to C), when we arrived back at that C we would be 41.06 cents flat... this is known as the "Diesis." Four minor thirds gives the "greater diesis," which would leave us sharp by 62.57 cents. There are many other commas... the "didymic comma," the "diaschisma"... you can research them. They all have names and were all studied and documented as composers, tuners, and instrument makers sought better solutions to the temperament question from the dawn of polyphony.
Eventually, we just kind of "gave up" and accepted the sour thirds of equal temperament because it makes modulation so damn convenient. But I do want to stress that the "circle of fifths" is not as simple and elegant as it's presented in first semester harmony classes... in fact, it's really nothing more than a fantasy that we've tried to will into existence, consequences be damned.
As you alluded to earlier, all of this is the reason why we have different names for the same pitch class. If we're in the key of C and have a secondary dominant II7, the F# in that chord is in actuality a VERY different pitch from the Gb in a C diminished that decorates tonic. On a piano keyboard, one key "stands" for both notes. But a violist with a good ear would play each differently!