It is not just temporary accommodation that can private, local authorities can also discharge their duty to rehouse a homeless household in the private sector. It is less common, but it is still possible and does happen.
However, it is important to point out that is not necessarily bad… If it is the right solution for the individual, it might help them get rehoused in an area of their choice.
What is important is that there are safe guards in place that the new tenancy meets their needs and adequate support is offered and delivered if required. That is where things can often go awry.