It could be changed, by reforming the welfare programs to stop making having welfare babies out of wedlock pay. There are a lot of possibilities for doing this, such as not giving out welfare benefits on a per child basis, not giving out services to the parent but directly to minors (a soup kitchen type setup). Also get rid of the Earned Income Tax Credit. The problem with the way that many of these programs work is they give more money to the parent or every illegitimate child, the parents can then use this money how they want, on themselves.
You make it sound like the biggest problem is that women sit down and say "You know what? I think I need a baby so I can screw the welfare system."
AFAICT, the actual problem is that not having control of reproduction leaves the women unable to implement long-term goals of not being on welfare. In which case there would be more impact from making access to long-term birth control like IUDs and implants cheap and easy.