Yes, but I think some parts of the Republican leadership view the abortion issue like Jesse Jackson et al. view racism: the problem is more valuable to them than would be a solution.
The point that politicians consistently miss, whether by design or blindness, is that the majority of the people in this country are opposed to abortion on demand in most cases, but a significant portion will fight tooth and nail to keep abortion legal in the few cases (e.g. rape and incest) where they think it should be.
Currently, such people are being pushed into the "pro-choice" camp by Republicans who take a hard-line all-or-nothing stance in a situation where it's really not tenable. If the Republicans could work to build points of agreement while "agreeing to disagree" on points of contention, they could use the abortion issue to sweep the Democrats off the map. Unfortunately, the combination of all-or-nothing-ness and wishy-washy-ness costs them a lot of votes while gaining them nothing.
You may be right. This seems increasingly to be the case with a growing number of third rail political issues, like immigration and Social Security.