'everything is a matter of good fingering. Hummel was the most knowledgeable on this subject.' also from The Method. Quite what's 'pivotal' about 1-3; 1-3; is quite beyond me anyhow.
Since Chopin is dead and all, we can't ask him, but if you analyze the "standard" fingering in this cadenza, then you will see that nobody has to stretch there, go into weird contortions of any kind, or change the fingering to something else because:
in the first group, 2 is the pivot and
in the second group, 3 is the pivot
and the pattern continues like that until the cadenza dies out.
So, although it looks like you are using the same units (1-3), their function is essentially different. Besides, in the second 1-3, you have the benefit of the hand that actually initiates the group according to Chopin's instructions - the slurs, that do not indicate "physical legato" but musical and technical groupings.
P.S.: Your "fourth finger" quote is evidence that Chopin thought that it COULDN'T be changed with any practice at all, just like the form of his nose. It's a very sarcastic remark, aimed at proponents of the "traditional" schools of his time with their crazy mechanical exercises to make that finger stronger.