Just my opinion, but I don't think "private markets" are incompatible with a proper valuation of human life?
It's a bit macabre, but, to me, "dead people don't pay taxes / generate GDP" seems pretty interchangeable with "dead people don't pay insurance premiums / contribute to a risk pool".
To me, the problem is that billed medical costs (what the hospital charges you) have come unhinged from realized medical costs (what the hospital spends on helping you, including salaries, supplies, etc).
If that world is alien-sounding, it's similar to how the billed education costs at universities have come unhinged from realized education costs. In both of those environments, the ballooning costs seem to be propped up by well-intentioned but fundamentally "unnatural" source of funds, in the form of government subsidies.
These have been exploited by profiteering and accounting optimizations, over the course of literal decades. If your surgery costs 2k to perform, the hospital will happily charge you 20k, collect 3k in insurance, and write off the remaining 17k as un-recovered medical expense, for which the government will reimburse some portion.
The more obvious solution to me, would be to take a hard look at how these subsidies are actually used (/gamed), and steadily reduce/focus them over time, so that hospitals don't have an incentive to inflate costs. The source of the problem isn't anything fundamental about "the private market", it's rather about the current set of public incentives that have distorted the private market into it's current form.