> The goal of a healthcare system ought to maximizing the health of the people.
Everybody disagrees with this. A more plausible statement would be that it ought to offer the best health/price trade-off. But even then only crazy fascists think that would be ideal. Go watch that Sliders episode where the CDC rules America.
> The goal of an educational system ought to be maximizing learning.
Really? Lots of people would disagree. You've avoided the question of what's worth learning, and I'd say it's far more important that an educational system instill a strong work ethic and a sense of morality that I can tolerate.
> Perhaps a free market system (used very loosely) with the goal of pursuing profits would, as a byproduct, maximize the health of the people or of learning.
You have created an impossible standard. No system you describe in three words is going to maximize your utility function, especially not one as absurd as yours. Also the free market vs. non-free market dichotomy is completely silly. With all the government funded research and state schools, you can't say that America's universities are a "free market". But they do have an important free market feature that lower-level public schools lack: choice of what school to attend.
> It is morally wrong for someone to profit from denying someone else healthcare.
I profit by not financing a tunnel through the mountain to the hospital. Sometimes an emergency vehicle takes too long to get to the hospital and this causes the patient to die. How is my behavior morally wrong?
> Indeed, in the case of healthcare I have very strong evidence that profit seeking isn't a great motivator.
No you don't.