First, the Hyde Amendment deals with public funding of abortions, not the legality of the abortions themselves.
I'm making some assumptions and what they think (the article does not explain the "why"), but the thing to keep in mind about statutory rape is that the sex in those cases is usually consensual in that both parties consented to the act. It is rape because the law has decided people under a certain age cannot consent
as a matter of law to having sex. The law has deemed them too young to be able to consent to the act in all instances. In Texas the magic age is 14. Anyone, and I do mean anyone under 14, cannot legally consent to sex, end of story. If I have a 14 and 13 year old have sex. It's rape. Period. It doesn't matter how often the 13 year old says they want it. They can say yes all day long. It's still rape.
There is the within three years rule, but that only applies with those who are between the ages of 17 and 14. If it's under 14. That rule does not apply.
Pro-lifers believe that from the moment of conception the baby is alive. To abort it is to kill a human through no fault of its own and is to kill someone who has no ability to protect their own existence. This is why many pro-lifers are as passionate as they are about the issue. They do not want their money being used to pay for act they believe is murder. The debate over whether a fetus is alive and when they are stems from this. Most pro-lifers are okay with abortion in cases of rape and incest is because the pregnancy is completely forced upon the woman in every sense of the term.
So why would statutory rape differ from forcible rape? In my example, both parties agreed to have sex and the 13 year old gets pregnant. As I said, this is rape. However, the initial pregnancy wasn't the result of an act to which they did not want to partake in. Whereas in forcible rape and most incest cases the pregnancy arose from an act to which he victim did not consent. If a person ardently believes that abortion is murder why would they ever want a dime of their tax dollars to fund an abortion that comes as the result of a consensual act? Statutory rape is still rape and is prosecutable as such, but in pure statutory rape cases the person who got pregnant said yes to the act that caused the pregnancy.
The Democrats are right in that the language could be interpreted in a way that excludes date rape, since that does not involve the use of force as it is typically understood. If I were a judge I'd never read it that way because that's not what is intended, but I can't guarantee others wouldn't. The other problem is statutory rape cases can be very complicated. What if for example it's two thirteen year old's? What if it's a thirteen year old and 8 year old and they 8 year old says yes? There is good reason the statutory limit exists (I think it primarily created to cut off the adult child predators defense that the kid said yes), but it's bright line also creates situations where you could prosecute a 13 year old for having sex with a thirteen year old. Actually, you could prosecute both of them. You'd probably have you ass kicked in front of jury for doing so, but you could legally charge both.
I don't think they have through all the possible implications. They are thinking primarily about the young teens and tweens who have consensual sex, but the ban would be broader than that. However, this is by no means unique Republicans. Law makers of all stripes do this all the time.
You can agree with the rational or not (and that language will probably cover more things than they ever mean it to), but I don't think their motivation is very hard to understand.
NOTE: All my examples include only juveniles because that is what I deal with in my job (you'd be surprised what some of these kids do), but such cases don't always have to involve two juveniles.