There are no free markets in human societies. You can pick your trade-offs, that's it. Privatly regulated markets can fail, goverment regulated markets can fail.
No, because this is always (ultimately) by consent of the societies it is happening in and within those power structures. It always sits on top of a lot of stuff, never in a vacuum.
There's an important distinction between observing the status quo and holding or aspiring towards ideals. If we only observed what is and ignored principles and ideals, we would never be able to improve the human condition.
Even so, I doubt we'll be able to see eye to eye given your starting premises about collectivism. I'd venture that the individual does own himself and that the liberal tradition starts from this point.
A consequentialist might venture that some of the darkest events in history flow from the collectivist ideologies. The opposing deontological view would be that the individual does own himself and private property does exist. We would struggle to define fraud and theft without these premises. Indeed, objecting to fraud and theft becomes problematic when you are simultaneously arguing that private property, individuals and free exchange are impossible.
I don't think we are that far apart. I don't disagree that there are certain ideals to aspire to.
However, practically, we find out time and time again that these ideals are not enough to sustain themselves and that in the defense of those ideals there are trade offs to be made that compromise them to some extent. Realising some idealised version would require different humans and I don't subscribe to ideas of realities that need those (in the same way that that speaks against idealist versions of collectivism).
The real debate is then what trade offs to make and there we might not agree but that is totally fine and part of the process.