I'm not advocating the cheapest service being the automatic choice, I'm simply advocating choice.
Even if we spend exactly the same amount on the services, the very existence of competition will make the services better, or faster or whatever we value most per pound.
That wasn't what I said. An inefficient company will fail. That's not the same as saying only inefficient companies fail.
That's not how business works.
My company is a very long way from the cheapest in our market, yet we still get a lot of work sent our way. I'd expect the same from the private sector providing public services - just allowing choice opens up the opportunity to choose a provider based on any measure of "better".
Equally your understanding of cost savings is flawed. Cheaper staff/raw material does not mean lower quality. I employ plenty of Polish workers on minimum wage who are twice as good as English staff on £10/HR. And close relationships with suppliers and careful purchasing decisions can increase VA without using inferior products.
As well as all of that, there are plenty of efficiency measures that the pressure of market competition can create that are restricted by the endlessly deep pockets of the public sector. Even in a fully mature market where all of the inherent cost savings have been found, the innovation found in the private sector can often invent an entirely new/better alternative, rather than just continuing to produce the same old services using the same old methods.