I agree that access to birth control is a strong factor and likely heavily confounded with women's education. I think there will have been examples of women's education but lack of birth control (e.g., predominantly Catholic nations, especially early on), not sure if there are many examples in the other direction though.
Unfamiliar with USSR -- is the birth rate high there despite lots of educated women?
I doubt it, because the xUSSR is an obvious counterexample.
Access to condoms is probably a bigger factor.