It's both, you're just misattributing what the deeper position is.
These guys are still "the hero of their own movie", they're not consciously "punishing women who have sex". They're pushing a narrative about "personal responsibility" first and foremost, possibly with some post hoc rationalizations about subsidization of negative behaviors.
Is this caught up in Biblical moralism? Sure! But without understanding the psychology of your opponent, how can you probe the weak points? How can we put the issue on a continuum and then come to some compromise between sides?
Example: Let me tell you, you're not flipping anyone on abortion. I've sat in a 20 week ultrasound with my wife, and logic be damned you'll never move me off the emotion state that abortion at that state is murderous enough to be morally wrong. I'm certainly not a hard-liner @ conception, but I'd set the cutoff pretty early, 12 weeks or so.
There's a ton of evidence that abstinence only education doesn't work (duh), and free childcare is way too big an economic ask to be realistic.
I'd say that free birth control (preferably as depo-provera or iud) is the most preferable (actionable) risk reduction mechanism, and I can't find a good conservative argument against it. I'd use that specifically as a litmus test of reasonability in the debate space. Of course, even if free, this will be underutilized (it is already).