This is, please forgive me, a very idealistic view of single parent families. It's very easy to earn too much money to qualify for welfare (or other government subsidies), while still not making quite enough to support a decent life.

There are frequently waiting lists, restrictions (my favorite is disability, where you are not allowed to work, but are given barely enough money for yourself to live on), and miles of red tape associated with government subsidies. There's also the very real cost to the pride of the entire family associated with these subsidies.

> why should the state allow for a child to remain in that home

Because moving a child is incredibly traumatic for the child and expensive for the state?