Turns out free access to birth control slashes the abortion rate. Less punishing women, less abortions. Cool, we've identified a point of differentiation.
Okay, so what did most of the 'pro-life' side go with? Shit, turns out they went with punishing women instead of fewer abortions and again and again and again. Well, that's not cool. For fairness' and balance's sake, I'll say that the pro-choice is probably less about integrity of body and more about wanting to fuck without consequence.
This isn't necessarily a good argument given that they have theological objections to birth control. This maybe indicates a general value which is an objection to technological modification of issues connected to reproduction as part of what may be a general reactionary attitude. This is consistent with for example, the early objections to IVF and the use of anesthesia in pregnancy. However, the second one of these could also be construed as a "punish women" goal, even as it has become uncommon. It might be noteworthy in this context that the IVF issue still is an issue for Catholic official doctrine but not almost any Protestants, and the objection to anesthesia in pregnancy is essentially gone completely. On the other hand, maybe looking at something more connected to male biology might help: if this is purely an objection to technical intervention in sex, then one would expect objections to Viagra and similar drugs. But they don't exist. So that's an argument against the technical intervention hypothesis.
Another possibility is that trying to understand is part of a general attempt to give broad explanations for what amounts to an attempt to modify old theology to handle modern technologies and dilemmas. Thus the exact results may be to some extent essentially stochastic. One example that might prove an interesting contrast in this context to the Christian right outlook is that of Orthodox Judaism. In some respects, Orthodox Judaism has more of an objection to birth control than it has to abortion. Fitting this sort of norm into any of the above hypotheses really seems like shoehorning.
This isn't necessarily a good argument given that they have theological objections to birth control.
Given that there's no god to specify what theology you get, this just raises the question — why do they have those theological objections? You're proposing what amounts to a null hypothesis in your notion that "the exact results may be to some extent essentially stochastic".
One example that might prove an interesting contrast in this context to the Christian right outlook is that of Orthodox Judaism.
Or the various cultures wherein are found ...
As Multiheaded added, "Personal is Political" stuff like gender relations, etc also may belong here.