Circle of 5ths and the 4-chord song mathematically speaking https://www.classicfm.com/discover-music/music-theory/four-chords-every-pop-song/
The way you worded your question sort of implies that only the last four chords of the circle of fifths progression would be used for a four chord progression. So, let's first list out that progression...
7 6 5 (4 3 2 1)
IV viio iii vi ii V I
iii are more than four steps back so supposedly they would be excluded. Keep in mind it a circle, there isn't a start or end, so maybe you're thinking of the section
ii V I IV.
There really isn't any math involved with this and I don't think the linke article does a good job of explain the music theory about this stuff.
Let's first list out the Axis of Awesome chords and the first four of Pachelbel's Canon...
I V vi IV
I V vi iii
First, neither of those is a segment of the circle of fifth progression.
The second thing is the notion that
iii is not used because of some connection to the circle of fifths progression don't really make sense when the article goes on to use Pachelbel's Canon as an example, because it actually uses the
The problem comes from trying to explain too much through the literal circle of fifths progression. Instead of that, a better fundamental concept to work with is roots by descending/ascending perfect fifth. When roots by descending/ascending perfect fifth are combined with the idea of harmonic sequences, it goes a long way to explain a lot of harmony, harmony that has been used for the last 500 years! Add in relative major/minor chords (which the article does mention) and you have the bulk of most chord progressions.
Using brackets to enclose the roots by fifths we can see those fundamental units and how they are repeated using harmonic sequencing...
|: [I V] vi [IV :|: I] ...
Considering the Axis of Awesome usually is repeated, we have
I V and
IV I as two ascending fifth progressions. This is literally a harmonic sequence, because the
I is elided, the sequence would be
[IV I][I V], but the
I isn't repeated so it's three ascending fifths
IV I V. The
vi can be called a deceptive progression (like the deceptive cadence), or it's the relative minor of
[I V][vi iii]
The Pachebel progression is very obviously built of ascending fifths using harmonic sequences. The is a common minor analog to that
[i V][♭iii ♭VII].
Other super common progression using this process are:
[I IV][V I] two descending fifths, sequenced by ascending fifth
[I V][♭VII IV] two ascending fifths, sequenced by descending whole step
[I V][VI I] two ascending fifths, sequenced by descending fifth
Notice that the last one is the common blues turnaround and it is just the progression/sequence inversion of
I IV V I. Some people get themselves all worked up over the retrogression of
V IV in the blues, but when you view progressions in terms of progressions by fifth and harmonic sequence, things all fall into place nicely.
You don't need to worry about chords to avoid. Try thinking positively about the fundamental relationships that underlie common progressions. Add to the stuff above borrowed chords and the tritone substitution/augemented sixth chords and you have a very nearly complete picture of tonal harmony. At that point you should have solid harmony and any supposed "avoid" chord, becomes either a potential source of surprise/uniqueness to exploit, or can be made to work through means like repetition, harmonic sequence, or modulation.