You forget the power of propaganda. Every time there is even a mention of any kind of "restriction" on abortion (even if it is just a common-sense measure like forcing abortionists to be truthful), the pro-abortion fanatics start screaming about how restrictions on abortion will cause contraceptives to become illegal. They also howl about how pro-lifers want women to remain barefoot and pregnant. Pro-life people who don't pay much attention to the issues hear the screeching, become convinced that the legality of contraceptives depends on abortion remaining legal, and for that reason support continued legal abortion.
Similar to OUR stand on the Second Amendment issue, with one important distinction. We can show historical precedent for our concern; they are simply lying.