I think a corollary of your title could be, "liberalization matters, not privatization". When most people argue for or against the "privatization" of health care, all they are really talking about is changing the payment stream so that individuals (out of pocket or privately insured) pay entirely for the same system for which the government now pays a majority share. But that does not involve changing occupational monopolies, disencumbering drug development, or making it possible for innovation to force prices down. That is flipping only one switch in a very large machine.
I think that this is a really good point. As a private individual I cannot purchase most medicines (even at the marginal cost) without a prescription from a medical doctor. But there are some conditions (e.g. hypertension) where I could probably do a better job of monitoring on my own using a snazzy blood pressure machine and being able to look at factors like time of day or day of week.
In the same sense, if I get an infection at 5 pm on a Friday afternoon, my options are essentially the emergency room or, just maybe, urgent care. There are no circumstances in which I could call a nurse with my symptoms and try a first line antibiotic before heading into the emergency room.
It is really unclear that market forces will work under these restricted conditions. And this ignores other factors like information asymmetry. Nor am I convinced that all of these conditions are necessary to force innovation forward -- restricted resources can also be an effective spur to innovation and American (and Canadian) medicine has mostly avoided that scenario so far.