Part A coverage is strictly limited to certain benefits, mainly for inpatient facility claims, but also including some home health and blood administration services, for example. But almost all professional services are covered under Medicare Part B, whether rendered in an inpatient or outpatient setting and whether employed by the hospital or not. If the patient has no Part B and no other insurance coverage, then the patient actually is self pay for physician services.
Usually if a patient is eligible for Part A but does not have Part B coverage, it is by their own choice as they have decided not to pay the Part B premiums to enroll. It doesn’t have anything to do with the patient not being able to choose who is contracted or on staff at the facility.