Do private hospitals have an incentive to cut costs? The government has a pretty massive incentive to cut costs. That's one reason healthcare in the UK can get so shoddy. Doctors are paid very little and they only increase funding when an outcry emerges. But it's very cheap. The UK's system is one of the cheapest there is.
If anything, private doctors have a pretty big incentive to charge for as many tests and services as possible. They get paid for each service, regardless of how much it benefited the patient.