If a parent cannot support their children, take them.
Leave the parent destitute.
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.
You are not that far off of the progressive plan to institutionalize all children and replace all families.