The circle of fifths progression (I - IV - vii° - iii - vi - ii - V - I in major / i - iv - VII - III - VI - ii° - V - i in minor) is extremely common in tonal music. Why?
If you - starting with a I7 chord you have 2 pairs of thirds (1,3,5,7) - descend in steps of diatonic seconds (first the upper pair thirds than the lower pair) you get exactly this progression. I assume this game was derived from playing the organ. I7-IV4/3-viiø7-iii4/3-vi7-ii4/3-V7-I:
1357 -> the 5 and 7 step down and become 4 and 6 -> 1346 = G7 -> C7 (second inversion)
1 and 3 - the lower third - steps down to 7246 and becomes the 1357 of the new chord = F7 -> Bm7b5 and so on ...
This game is also functioning with chromatic chords and secondary dominants along the circle of fifths.
That’s why - in my assumption - early Baroque composers (Vivaldi, Corelli, Buxtehude et al) used this “tool” to establish a tonal center and spinning on going through the harmonies (continued by Bach and Händel, and the Classical Epoch till pop music of today):
Cmaj7-Fmaj7/C-Bm7b5-Em7/B-Am7-Dm7/A-G7-Cmaj7/G ... etc.
Go from somewhere on the circle to the next, clockwise, and you've moved a fifth. As in, the first chord/note is the dominant of the next. And so it continues. The most common harmonic move is V>I, or more pointedly, V7>I, as the tritone produced sounds like it needs to resolve itself at the I. It's that tension/resolution feel that music, at least in the Western world, thrives on.
You asked a very similar question before.
I think you should be careful to distinguish between statements of the entire circle of fifths sequence versus shorter segments of root progressions by descending fifths.
The full circle of fifths sequence is common, but maybe not as common as you are suggesting. By comparison harmony by descending fifth is super common, but such progressions are not always sequential.
If you overlook the fact that the full circle of fifths is actually four sequences of root progression by descending fifth you will miss the important sequential character of the full circle.
The full circle is root progression by desc 5th sequenced down by step 4 times
[I6 IV][vii6 iii][vi6 ii][V6 I].
Simply change the sequential step to upward and we get another common sequence nick named the "monte"
[I6 IV][V6/V V][V6/vi vi].
My point is to sort of warn against too much attention given to the full circle of fifths instead of recognizing common sequential patterns and see the full circle as one member of the family of common sequences.
Similar patterns can be found with harmony by roots by descending fourths. "Falling thirds" and the "monte romanesca" are common varieties. That last nick name was coined by Robert Gjerdingen in his book Music in the Galant Style.
I really think you want to combine an understanding of this kind of sequential harmony with the rule of the octave which essentially codifies various
I V and
V I movements. This is actually how composers like Mozart, Vivaldi, etc. learned and taught harmony.
Sequences of seventh chords by descending 5th appear in the work of Baroque composers like Corelli as the fruit of a quest for a limitless succession of 7th chords.
In the late Renaissance (16th century), as everyone knows, the basic chord was the triad. Seventh chords could be used if the 7th was treated in the manner of one of the non-chord tones, e.g. the suspension: e.g. in the cadence
vi - ii6/5 - V - I
(the functional labels are of course used anachronistically), the 7th of ii6/5 is prepared by the 3rd of vi and resolved (downward, as with all suspensions at the time) into the 3rd of V.
In the 17th century, composers discovered that they could link 7th chords in this way, each 7th prepared by the 3rd of the preceding chord and resolved into the 3rd of the next, producing a sequence by descending 5ths.
Eventually the progression became so common that the 7ths could be omitted from some or all of the chords and still have the same predictable effect.
Note: it is also possible to resolve the 7th into the root of the next 7th chord, producing the other staple sequential progression of functional harmony, descending thirds (I7 - vi6/5 - IV4/3 - ii4/2..., or just I - vi - IV - ii...).