If we are concerned with nuance, there's probably at least 2 different conservative arguments.
There is the hyper-libertarian argument which says that if someone arrives at a hospital unconcious, and the hospital bills them the maximum amount they estimate the patient can pay before bankruptcy, so the hospital can maximize initially reported earnings before writing down unpaid debts, that this is somehow a voluntarily market price as long as the hospital is private and not owned by the government, because value is purely subjective and has no relation to cost.
Then there is the other argument which says if hospitals are over-billing people which are under duress, that this is bad and we want to do something about it, but we don't want to immediately nationalize healthcare and ban private health insurance. This second type of conservatism usually focuses on price controls or regulation of monopolies.
With pricing transparency regulations, it's probably possible to implement universal public price negotiation by fining non-elective healthcare providers which bill patients over 100% of their previously published price or 120% of the minimum price published by comparable providers. Then rebate patients the entire amount they were over-billed regardless of their insurance status.
With over-billing rebates there is less need to collect payroll tax to finance social insurance premiums. Need based assistance could be financed from more progressive property taxes on the rich, and the actual price controls could be implemented relatively cheaply by randomly auditing providers to ensure compliance, by allowing patients to manually submit invoices whenever they felt they were over-billed, and by financing rebates to patients entirely from fines on providers.
It's certainly possible that other systems would work better. However if we want to be nuanced there are many different strands of U.S. conservatism. Historically U.S. conservatives subscribed to some form of cost or labor theory of value. The kind of hyper-libertarian conservative movement which says private prices are always fair is probably less than 100 years old.