Why is it that a baby born on Shabbos via C-Section cannot have a bris 8 days later [on Shabbos]?
Rambam Laws of Circumcision 1:11 codifies that the bris of a boy born via c-section does not override Shabbos.
If I recall correctly the source of this correctly, the Torah's special exception to allow circumcision on Shabbos (as any other elective surgery would be prohibited, and circumcision on any day other than 8 is prohibited) applies only to a boy who is "birthed." (Leviticus 12:2, Isha ki tazria, v'yal'da zachar.)
There is even an opinion that a boy born due to artificial insemination does not fit the category of "ki tazria" (should a woman conceive), and thus his bris would also not override shabbos.
As far as what meaning we could draw out from such a law, I'd assume the idea is that only "wholly conventional" births, on day eight, override shabbos, as a way of stressing that shabbos isn't overridden willy-nilly. But that's just my guess.