As you have invited personal criticism: You are naive and your thinking is simplistic. You have managed to get away from complete naivete, you think using abstract concepts, which allows you to look through the superficial appearance of things, and thus cut through a lot of bullshit. But you haven't noticed yet that your abstractions are way too abstract to give a reliable representation of the real world by themselves, that abstractions are just helpful thinking tools, but not the full truth - and that because of that your way of thinking produces about as much bullshit as it cuts through, just of a different kind.
In particular, it's seldom helpful to cling to specific terms and their supposed definitions rather than look at the actual facts. Who cares whether doctors are slaves? Is there anyone who has any actual disadvantages due to how the system works or not? If everyone, including the doctors, is better off with how the system works, who cares whether they are slaves by some definition? If they have some disadvantages, then that should be addressed, who cares whether it fulfills the criteria for being a slave?
Also, you always have to consider the wider societal context. There might be indirect benefits outside the immediate transactions that arise from how a society is constructed. Where I live, for example, if you see someone in distress and there is no significant danger to you if you were to rescue them, and you are able to, it is a criminal offense not to help. I guess by your standards that would mean we are all slaves - but I think the democracy that created the law is perfectly sufficient to justify this obligation, which as a side effect presumably generates higher safety and a higher sense of safety for everyone.