Profit is a structural driver for efficiency. Poor efficiency also means less money for the actual service.
If you can go, for example, from 20% inefficiency to 10% inefficiency and 5% profit, the service has still gained. Aiming to get 10% efficiency improvement without a structural incentive is unlikely to happen because there is no gain.
Is "efficiency" a sensible target for healthcare? What does it actually mean in a healthcare context? Markets work very well in some areas: specifically where we can have real competition, easily measured outcomes and good customer assessment of what they want. None of these things are true of healthcare. Patients are extremely poorly placed to judge the quality of healthcare they receive or will receive from different suppliers.
In any case, the fact is that the NHS performs extremely well on efficiency measures by international standards, while the developed world's most free market, private, system - the US - does extremely poorly so why would we think that introducing private competition is going to help?