In fairness Blair introduced the PCT model, primary care trusts.. they had very strict waiting times and if the patient was later than 18 weeks being treated then the PCT had to find the operation somewhere, either at another trust or private, as they were responsible they took the choice to do private. At the time Dobson went nuts, but as PCT were independent he could do nothing. So blair was responsible from a bit but mostly it was the boards of the PCT, most run by tories, it was they actually introduced the private sector to the general NHS, all for political point scoring.
Of course large swaths of the NHS have been private for years, Dentist, opticians, pharmacy, chiropody etc etc