The idea that government can offload social programs onto churches is based on a complete misunderstanding of what churches are.
A church is an institution in which people worship God. The point of a church is to provide a place for a community of worshippers to engage in the rituals of their faith.
A church is not a social service institution.
Although most churches engage in some charitable work, that is not their main purpose.
There is, in the eddies of Christianity, something called the Social Gospel. But that is not the majority position among Christians. Christians are not bound by that bit of philosophy.
If society has an obligation to meet the needs of the poor, that obligation must be fulfilled by the central organizing institution in that society: the government.