As for the latter, I don't know if this is a "logical" basis or not, but I think it's based in part on the idea that there has to be a specific defined moment when human existence begins. And it's much harder to articulate that defined moment as lying on a spectrum, rather than one of the two categorical changes in human reproduction: birth and conception. Both of those changes happen in a near instant, while other developmental events happen in a gradient. And since many (most?) people feel that an hour before full-term birth a fetus is actually a human baby, the fact that there's no single instant of category change before that will lead some (many?) people to search back to the last category change as when that switch from not-a-human to human baby happened.

I agree with the first part, that they think sex is reserved for procreation, and that anything else is immoral, but there's a more cynical view of that latter.

The idea that a person is made when the sperm enters the egg became a tenet for non Catholics when Evangelical preachers discovered it filled coffers. 100 to 150 years ago, preachers actually advised young women to get abortions before the quickening. So the cynical view is that adopting the moralistically pious attitude that the sperm-egg created a human and the soul entered at that point worked better at getting money, retaining andd expanding followers, and religionists who adopted this survived much better. And no one worried. If you were prosperous, you could always get an abortion, it was more difficult for the poor.

But these people want to make laws so that if you get your abortion out of state and return, you are still guilty of infanticide. They are somehow fanatical in their desire to punish sex.