Because there's a large subset of people that believe God wants all children to live, and simultaneously believe once they're out, they're someone else's problem. They never seem to connect the two.
Before making a 'qualified' statement, get your facts straight first.
Infants don't have personal responsibility, so cutting programs that benefits children hurts them more than the parents.
The same thing goes for benefits programs -- nobody wants kids to be starving in the streets, but we would rather that people exercise personal responsibility in taking care of their own kids rather than expecting the rest of us to pay for their care.
Those positions are not opposed, and there's a lot of nuance in the whole welfare debate, it's not all or nothing.