Now, if you are only using 3 chords that are in a circle of fifths progression, let's say G to C to F, then that does not really qualify as a circle of fifths progression. You really need to use at least 4 chords, such as G to C to F to A#, to qualify as an official circle of fifths progression, although that is not a hard and fast rule.
Any functional V->I progression counts as a "circle of fifths progression", even if it's only two chords.
Since your example progression it's long enough to establish C as the tonal center, the
best explanation is a circle of fifths progression.
If you extended the progression long enough to establish C as the tonic, the
G C would still be a circle of fifths progression.
Personally, I've always explained the circle of fifths to my students in terms of the
harmonic series, since it gives a good explanation of functional consonance and dissonance. It's also worth noting that in that context, the "circle of fifths" doesn't actually return to the starting note, since there's a gap known as the
Pyhagorean comma, which then leads to useful discussion about tempered tuning.
The importance of the circle of fifths progression is that it can make your music sound more interesting. The reason it is making your music sound more interesting is because you are frequently modulating to a different key.
Plenty of circle of fifths progressions don't make use of modulation, and plenty of modulations don't make use of
tonicization. Most do, because the V->I progression is perhaps the strongest progression in Western music, so it's often used to clarify to the listener that there's been modulation... But there's no
requirement that the V->I movement be involved. For example:
D | Dm | Em | Fmaj7 | G | A | Bm | C#dim | D Here, I've temporarily modulated from D major to D Dorian (or perhaps C Ionian, but because I wrote it I'll claim it's Dorian), moved in a stepwise progression, and then modulated out back to D major again, with nary a V->I motion in sight.
Of course, if you are interested in writing top 40 style songs, or songs for film, tv, or ads, you should never use the circle of fifths, because, in those styles of music, they are just looking for musically boring songs that sound similar to songs the listeners have heard before.
Because V->I progressions
never appear in popular music, with the exception of just about every pop song ever written. And temporary modulations
never happen in popular music, except
all the time.
To sum up, the circle of fifths is merely a chord progression in which every chord is 5 semitones higher than the previous chord and it can be used anywhere in a song, verse, chorus or bridge, and it will make your song sound more interesting and more musically sophisticated.
Again, you appear to be talking about temporary modulation using a tonic pivot chord, not the circle of fifths.