My fear of robot doctors are ones fully trained and controlled by the company creating them or insurance companies.
With a human doctor there may be variability in how they treat you and what they tell you - you could appeal to a person's empathy.
But what if robot doctors are trained by insurance companies to withhold information or procedures that should be done if the patient doesn't have insurance or can't afford it?
Human doctors do this too - I'm not denying that, but at least there's some variability there.
With a robot doctor you could have a 100% success rate in withholding information.
I could be entirely wrong, I don't know any laws surrounding withholding info, but there could be some legal grayness there.
An insurance company would never admit that their robot doctor was trained to withhold info. They could just say that their robot doctor got it wrong.
[EDIT] This is definitely a people controlling an AI problem versus an AI problem, but I still think it's worth calling out. Maybe it's the pessimistic inside of me, but I just don't see a future where specialized AI X will be trained in good faith by companies.