I don't think that there was a process of anyone "allowing" rehoming as much as it having sprung up without any authorities, or the people who make laws, being aware of it. The internet has taken a practice that was long used informally within families to allow family members to ensure that children are cared for (grandparent custody and the like) and blew it up to ridiculous proportions.
But, I think it has also been fed by a couple of other factors. One is the growth of international adoptions and a lack of oversight. Some families go this route to get kids that they believe are whiter and less likely to have behavioral/medical issues. In some cases other countries have played into this, and in other cases dubious agencies have provided connections. Another wrinkle to the international adoption field has been "quivering" families, like the Duggars, whose religious beliefs lead them to a belief that more is better in terms of children and that they are to raise up armies for the Lord. Many such families have gone to countries with lax adoption regulation, returning home with multiple adoptees.
In all of these cases (as well as the Harrises who apparently pushed their way through to adoption of some kids with serious histories of abuse) families are ill-prepared and poorly supported. Many who came in with the notion that love, or discipline would magically make all the hurts go away, have been sadly disappointed and quickly overwhelmed. Some children have been disciplined to death.
As an adoptive parent, I am fully willing to concede that the authorities are clumsy at best when it comes to placements, determinations, preparations and ongoing support. But, they are the best we have, and we need to work at shoring up their legal authority and resources.