No, I'm not talking about rising costs in terms of the UK US comparison. I'm talking about total cost per head of population of the model. The US model of healthcare costs more, per capita, than does the UK version, yet the UK version delivers a free at the point of need service and the US version does not, meaning millions are excluded from healthcare in the US whereas everyone gets care in the UK.
The best healthcare system in the world, according to those who know, is, I believe, France, which is almost totally state-funded too.
Socialism requires that the state owns everything. In both the US and the UK, the state owns very little, certainly compared with most of Europe. However, in both of those countries, most things are owned by a small number of highly wealthy individuals who would all argue that they got where they are by merit. I assume you'd accept that as reasonable.
Hayek is hardly a source of reliable economics. I've not read anything of his that made any kind of logical sense.
Could you give me some examples of the historical evidence of which you speak? I have seen some evidence, both historically and contemporaneously, that competition can drive down cost and increase quality. I have also seen the precise reverse. As I'm sure you're aware, asking me to prove the reverse of your statement is somewhat of a fallacy, so where is your evidence in support? Historically, humans have lived in small units (under 250 people) and cooperated for success. I'd love to know what the compelling historical evidence is that humans work better under conditions of hard competition, please post it :)

In fact, I'd love to post a full response to not just your post, but a number I've read over the past few weeks, so if you're up for this, can we continue this debate here, but once I've put a little flesh on the bones of my arguments (and found a couple of pretty pictures to make it appealing to steemit!) I'll tag you in a post? I love robust debate, so would greatly value your input on my future post too :)