The point of such discussion is about respect.
You pulled someone up for not knowing what they were talking about ("For someone who puts so much faith in free markets you don't appear to understand how they work. In your studies, look up "market elasticity" for a clue."), when it seems the same accusation could be levelled at you.
But, I never really meant it to become a big deal so I apologise for being picky and I'll move on to answering your question.
Broadly I agree with you but I think it's slightly more complex than that a straight forward "it doesn't work" and for me the issue is how you factor in the failures which in healthcare tend to result in injury or death. For instance, would a law covering corporate homicide (that is an organisation causing the death of someone in the course of their work due to incompetence or carelessness) count as market regulation? Or even manslaughter if a doctor acting alone did the same? Would that be regulation?
Because of this I think any civilized society wouldn't accept the possibility of severe injury or death without significant consequence, so even if it might work in theory (more below), we'll never know in practice as it would never be allowed to come about.
More generally though my view is that markets of any sort only work well when you've got a few factors at play:
(1) the deals and decisions are being made directly by those they impact - that is the patients so the deal genuinely reflects the will and interest of the patients (as opposed to them being one step removed as with insurance companies where their interests muddy things)
(2) the system has to be such that there is a financial consequence to the patient of the decision they make (that is they see the benefit of getting a deal which is good value as well as good medicine otherwise why would they look for value)
(3) there has to be as close to perfect information as possible to allow these decisions to be made on a true and fair basis - and when you look at the PR machines we see today exist, you've got to think that's not going to happen without regulation.
My issue is I don't see how you'd consistently get those factors without regulation. I couldn't say it's impossible but it certainly seems highly improbable, certainly in the culture we have now (point 3 being the one I can't ever see coming about without enforcement).
As an aside - I live in the UK where we have nationalised healthcare and it's frankly great. The market reforms over here terrify me because they're ideologically driven by people who seem to want to drive us towards an American system so in theory while I'm not intrinsically against the market playing a greater role in health provision, in practice I'll tend to fight it on the grounds that it looks like they're going to the whole thing up a treat.