One of the reasons a "well-designed social service/institution" will not work as well as a religious institution is that we have a natural bias from childbirth toward religious belief, no matter what the indoctrination. The specific research summary I am thinking of that justifies this is Justin L. Barrett's
Born Believers: The Science of Children’s Religious Beliefs.
So, supposing an alternate institution were designed and everyone were required to participate, in time, and probably very quickly, people would re-create religious institutions, underground, if necessary.