The CMA published its market study into children‘s social care in England, Scotland and Wales recently, it’s an interesting read, and found that whilst profits in private sector facilities are higher than might be expected in a well functioning market, the actual fees they charge to councils are not materially higher than the cost to councils of providing their own placements: www.gov.uk/government/publications/childrens-social-care-market-study-final-report
The crux of the matter is that it’s not a well functioning market and doesn’t meet the needs of vulnerable children because there aren’t enough providers overall, there aren’t enough providers in the areas they’re most needed, and there particularly aren’t enough providers who can take on challenging children and those with the most complex needs.
Yes, councils sometimes do have to send children to “whoever will take them on”, because of inadequate provision, largely driven by a lack of centralised planning.