I think you hit the nail on the head. Once abortion was out there, they didn't need it, and didn't need to disturb some of the more squeamish sensibilities.
The progressives were the ones who started eugenics and sterilization, but now they blame it on everyone else?
Mind you that in every state poor women can still get sterilization for free if they ask or request it. Problem is the abortion industry has their claws into poor women, that will pay for an average of 5 abortions in her life time.