The argument given is its meant to set an example, but in reality it is a desire for a form of frontier justice.
The same mentality drives the Texas opinion on the death penalty. No matter how many times I argue with a true Texan that the death penalty doesn't have a good track record for disincentivizing crime, or even the fact that it is more expensive than life in prison, the support for the practice never wavers. In their answer you can hear the desire for wild west-style justice, from the often said comment of "it shouldn't cost more, a rope/bullet is cheap" to the notion that despite the extra expense (or even the fact that innocents have been executed) the punishment is worth it because some crimes don't deserve the leniency of modern civilized society.
A true Texan will be more charitable to children than many in this thread will admit though. Many of the Medicaid Obamacare extensions that were actually allowed in the state via carve outs all involve children. Texas also has a "robin hood" system to subsidize poor school districts via rich ones, has a massive free lunch and breakfast program, and guarantees acceptance to any student to any public Texas university within the top 10% of their high school (a huge boost for those who come from less reputable high schools). I don't think Texas avoids caring for children when it can, as the children are sympathetic as in its easy to feel bad for them.
What is different in Texas is the outright scorn and disdain for the parents of said children, they are given nothing and trusted with nothing. Many of the benefits in Texas (like the free lunch program) exist as a way to not give welfare money directly to parents who will choose to spend it on drugs or booze. There are no GED advertisements promising the person who got pregnant young a better life for THEMSELVES, they are an untrustworthy deviant who must pay for their mistakes. This is where the Planned Parenthood part comes in- any notion of abortion or even subsidized women's care all focuses on what is best for the parent aka a person with no value or consideration in Texas. The times when children are hurt by this mentality are all accidental (though unfortunately often), the real target is the person who didn't do right by society and is lucky we still don't hang people from trees for the slightest transgression.