The NHS has always included ‘private’ elements - notably most GP practices. There are however big differences, for example in contract forms, employment and pension status, asset ownership, etc - between this (which was always similar to the French system) and the Thatcherite, selling-off-the-family-silver to mega-corporations kind of privatisation the Tories favour.
(I know this stuff, by the way - it was part of my professional life - I was for example the advisor that moved a thousand NHS staff out of Hull PCT to the City Healthcare Partnership Community Interest Company - among other similar jobs - all done with trade union and staff backing precisely to avoid Tory-style privatisation.)
In utilities, transport, health, education, and other areas, France uses private providers, but generally with the state retaining core asset values and control, and often employer status too. It has not been dominated by that naive Tory ‘market knows best’ ideology.
On your ‘deadwood management’ point by the way - see my post: