As the welfare state has grown, people have become less inclined to support private organizations and religious groups devoted to serving single mothers. This has not been due to any lack of generosity. They have paid for Aid to Dependent Mothers with Children and the WIC program with their tax dollars, so they do not believe further assistance is necessary. Whatever the imperfections of private charity, it is superior to public welfare for many reasons. It is voluntary, reflective of genuine charity, and more often infused with some level of moral concern.
Private charity can encourage sexual responsibility - even as material assistance is provided.
Surely abortion opponents can see that this is unwise public policy. Instead, we should hope to stop having the government attempt to centrally plan the lives of poor people. This alone would cause old-fashioned morality to undergo a sudden new resurgence. That's a result everyone can live with.