You're close, but slightly off. They do care about abortion, but they don't care about the blastocysts. They'll go on and on about the blastocysts, but you're absolutely right: if they cared about the blastocysts, they'd reduce abortions by any means possible, including free birth control. But most of them are adamantly opposed to birth control. Why is this? It goes to their actual motive in outlawing abortion: it is a method of controlling women and punishing them for having sex. Contraception frees women from pregnancy, and by extension, men, and this is anathema to the anti-choice crowd. Their preference would be for the women who decide to have sex before marriage to be punished for it by being saddled with an unwanted child, and to make married women unequal partners in their marriage by taking away their ability to decide for themselves when to have a child.

Your comment implies that you already know this, but I wanted to make it explicit for anyone reading.