The only reasons a “parent” could not support his/her child would be apathy (mostly due to selfishness or ideology or a combination of such), destitution due to the state, or state micromanaging in the affairs of the family.
Creating public boarding schools is not “eliminating all welfare” but merely transferring it from one outlet to another. And the state has no right to take kids either way.
Any parent that does not support their child is abusing/neglecting the child.
That's prima facie cause to remove the child to an institution that can meet their basic needs.
And it would be no more expensive, possibly LESS expensive that Welfare...which supports the piece of sh!t parent too.
It would also sever as a disincentive to combining poverty and pregnancy...ultimately reducing the number of children in need.