I think you and I probably would have different notions of what it means to "solve a problem." In this case I suspect you have a notion that a free market in many ways nicely sidesteps needing to even resolve this question in the first place. It frees one from having to commit to overarching, centralized value systems and instead allows for gestalt value systems to arise naturally from the behavior of people. This nicely avoids the issue of needing to impose a higher authority's will on a population and all the authoritarianism that that entails as well as the inevitable schism between a centralized value system and what people actually want. (I happen to disagree with this take and can expand on why if you're curious, but if this accurately reflects your views, there's enough commonality at least for me to make the next point.)
> increase the set of choices available to the person
This view abstracts behavior into that which is governed by "choice" and "coercion." I think this binary distinction is a fine model for a lot of domains, but a poor one for healthcare.
Choice feels much more like a spectrum in the domain of healthcare than it does in other domains. For a rough sample of points along this spectrum, you have "do this or die immediately," "do this or die in the next several months," "do this or suffer permanent disability," "do this or suffer great pain," "do this or suffer some probability of some amount of disability," "do this or suffer mild discomfort," "do this or be slightly annoyed."
The far-"left" part of this spectrum cannot ever realistically expand its set of choices. The most extreme version of this is that you're literally incapacitated and so can never make a choice of e.g. what hospital to go to and what treatment to administer no matter how many hospitals or treatments exist.
However, I think the same problem persists in less extreme states as well. Health ailments can directly impact a person's ability to choose to begin with in a variety of ways apart from just physical or mental incapacitation or degradation. Various treatments and healthcare choices impose switching costs that reduce a person's choice even when they are nominally capable of making one. For example, if a patient chooses a single hospital for a bout of appendicitis (when they are in such pain that they cannot make a choice in that moment other than to dial 911), even once the acute problem of surgery passes, they are unlikely to be able to choose a separate hospital for their post-surgery hospital stay without jeopardizing their health due to movement and continuity of care concerns.
Even in non-emergency cases there is an extreme information asymmetry and unpredictable path dependence (certain choices lock into other choices down the line but the nature of how they lock in may not be apparent at the beginning) that make it hard to formulate what "choice" would even look like.
In some ways, I personally view the need for coercion in the healthcare space as precisely a way to return to a world where modelling things as a binary distinction of "choice" vs "coercion" makes sense again.
Any plan for regulation of healthcare always must deal with a distinction between "elective" and "necessary," "non-essential" and "essential," "covered" and "not covered." That line is drawn precisely where we have a best guess that the model of a binary "choice" vs "coercion" holds vs the model of a spectrum of choice; the ideal is that care is provided to boost a patient back into a universe where the binary "choice" model is a reasonably good approximation.
More generally there is the problem that healthcare has a weird squeeze of monopolistic and non-monopolistic needs.
At a base level, in almost all domains including healthcare you need some amount of a regulatory framework to counteract the problem that market participants generally have an incentive to decrease the number of choices to the other side. I think you probably agree with "coercion" at this level (stuff like preventing collusion among players, certain stances on breaking up certain kinds of monopolies, etc.).
But the problem is that in healthcare you do want powerful players because there are benefits we want to reap from large players. Large drug makers are the only ones capable of performing substantial R&D and regional hospital and transportation networks are really the only ways you can get the necessary infrastructure and expertise to treat a lot of things. On the buyer side you want large insurance pools to even out risk for people.
But those all have inherent monopolistic tendencies that are exacerbated by the problems of choice that I mentioned.