Brit on Shabbat

Why can’t a child born by C-section on Shabbat have his Brit on the following Shabbat?

The Torah in Vayikra 12; 2-3 writes "When a woman conceives and gives birth to a male she will be contaminated…. On the eighth day his foreskin shall be circumcised" from this verse and it's juxtaposition Rabbi Chama learns in the Talmud, Shabbat 135a that only when the mother is ritually impure of the birth Tumah then the boy will circumcised on the eighth day even on Shabbat, the Rishonim dispute whether the Halacha is like Rabbi Chama and because of the doubt – we circumcise on the eighth day but not on Shabbat. A C-section baby does not render his mother ritually impure of the birth Tumah.