The current system - a bunch of clinical commissioning groups buying care from GPs employed by their surgeries not "The NHS" and from a bunch of separate hospital and mental health trusts not "The NHS" is pretty much what you describe.
We see a lot of private provision via the NHS (NHS patients going into private hospitals) in mental healthcare, and it's almost all lousy. (Even worse than the NHS MH care, and that's saying something.)
When you have 1 million patient contacts every 36 hours the gap between "good enough" and "good" is a lot of people and a lot of harm.
There's a lot of talk about privatisation of the NHS, but personally I'm really not ideologically attached to the idea of a state-owned health service provided coverage is still universal and free at the point of use.