Yes they are but they are indeed service providers as you mention.

That's quite different from a private practice (like a solicitor here) that you pay directly and/or that seeks payment from health insurance.

There are a great many things that the NHS pays practices for on a unit basis which is very much like them seeking payment from an insurer. The system has a far lower administrative cost than the USA model but the contract management process still looks more like a plate of spaghetti and not a circuit diagram.