You've just moved the problem to explaining why "the percentage of the population taking birth control has increased significantly since that time".
Your initial claim was that it was the availability of birth control. That is, that birth control was the cause. But this new claim is that it is the choice to use birth control. That is, choosing to use birth control is now an effect, which must have some other cause.
Cultural shifts, in this case due to the availability of birth control, take time to materialize.