This shows a total misunderstanding of economics.
The question is what those returns are, and who they are for. Private companies are, by definition in a free market system, looking to make a profit. They will therefore (in exchange for participation in the free market) demand a share of the returns from the allocation of resources.
When you speak of "returns" it sounds like you are talking about the profits for the provider. But there is another side to the "returns". The product or service. The provider will make money, and they should. They earn that money by providing the service, so half of the "returns" of a free market system are in the form of the product or service provided to the consumer. This is in fact the main "return" that we should be concerned about. The provider only gets their "return" as a result of being attractive to the consumer, meaning they must be competitive on price and quality.
The way you seem to be looking at this is that somehow if someone makes a profit, that would take money out of the sector. So when government spends money on healthcare, does anyone profit? contracts are handed out, and the government pays for them. They tend to be quite expensive too.
The first fundamental question is to what degree that share of returns taken out of the system by the private company counteracts the benefits of a more efficient allocation of resources.
You also seem to think that the efficient allocation of resources is secondary to "returns" being "taken out of the system". Do you not see that the efficient allocation of resources is what will achieve the lowest cost and the best quality?
And that's without even considering the question of 'fairness' (which I suspect that you'd say shouldn't be considered).
Yes that's right, because I advocate a free market I do not care about people and have shares in a private medical company. The fairest thing for everyone would be to let people make their own economic decisions, and to let the market do what men with good intentions and grand ideas cannot do.
You can only make economic decisions if you have the funds available to do so. That was why the NHS was created in the first place.
As for being an advocate of a totally free market are you also against government regulation of that free market?