Capitalist, yes. Free market? You seem an intelligent dude, so I'm not going to behead you for that ignorant remark. The only free markets in the US, and most developed nations, are referred to as "black markets" by the media. And they are indeed prosperous. The rest is regulated unto death. A great deal of the lack of labor participation in the US is due to the fact that if you try to engage in any sort of commerce without a small to large fortune backing your endeavors, the barrier to entry via licensing, inspections, necessary bribes, ad nauseam make it nearly impossible. In a free market, the failure rate would be higher, but the losses generally smaller, as the barrier to entry is essentially skill and guts.
"Free-market" is very abstract term. This is mostly true if you speak about trade barriers (during last decades most of them have been removed, allowing corporations to outsource labor to third-world countries that affected wages and employment the same way as automation).
Moreover, removing entry barriers and licensing won't significantly affect employment. It will simply increase competition in the currently-protected sectors slashing the wages and profit margins. For example, if the government will entirely remove all taxi regulations, some unemployed will get opportunity to earn living with driving own car while many professional taxi drivers will be pushed away from the market with falling fares.
Also, of note in your own reply, yes the most developed nations are capitalist. Funny how that works, isn't it? I'm not speaking theoretically when I say socialist societies eat themselves. I'm old enough to have observed it, multiple times. The Soviet Union achieved rather more than most of them, but in the end the necessary contradictions of "public" ownership, as opposed to individual ownership, led to their demise. Socialism is a beautiful concept, if you don't look too deep. It works great for ants, bees, and the Borg. But humans are NOT faceless, replaceable parts in a giant machine. Trying to make us be leads to failure.
There are simply no capitalist solutions for the technological unemployment problem. Period! Even moderate right-wingers accept this fact (you cannot call the author of the book "Lights in the tunnel" as socialist because he fully advocates market-based economic model, but nevertheless he insists to implement unconditional income TO SAVE this system). If insane right-wing fanatics (they call themselves "neoliberals" or "neoconservatives") won't change their position, they all will be killed during revolutions.
For one thing, technological advancement does not happen all at once and all in one place. Thus when a certain threshold is passed in one place, those in others observe, react, and where necessary, change course. In a truly free market, this is a quick, dynamic process. In what passes for markets now, it isn't.
World elites now mostly consist of insane neoliberals who try to hide any problems and keep status quo until the end.
Free market is not an abstract term, but it has been deliberately obscured for a long time. Primarily by the followers of the "great" John Maynard Keynes. He is one of the people that makes me wish I believed in an afterlife, because the thought of that son of a bitch rotting in hell for all eternity makes me feel all warm and fuzzy inside.
For perspective, I'm coming from the Austrian school of economics, which has thus far been very consistently correct in it's predictive and descriptive ability. Keynesian economics have also been very consistent. It hasn't been right once.
However, I completely agree with your last sentence. It's a large part of why I find cryptocurrency so attractive. It's an attempted end run on the system. I don't know that it will work in the long term, but it's making them nervous. I like it when they are nervous.
I am a free market anarchist. I developed my own philosophy over several decades, so I can't point to one school of thought that defines my philosophy, but the closest would be agorism. I don't see competition as a bad thing, and in your example of taxi drivers, the best service would win out. Which is a function of both price and service. Absent the regulation, it would tend to the mean. Yes, some would win and some would lose. To say that regulation and barriers to entry have little effect on employment is frankly to ignore history completely. Even the most diehard centralist is well aware of the direct correlation there. However, "unemployment" is in itself a somewhat misleading term. I assumed it to mean, in your context, gainful work. If you mean gainful work
in the service of another entity, then we have been arguing to cross purposes anyway.
I maintain that we have not run out of things to do, and for gain, and that we never will. The robots are tools. Who's hands they fall into will make a lot of difference in the outcome, but people will find a way to subsist.
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In a previous post I said that extreme automation is feasible but this does not mean that is economically viable... in the free-market capitalism. You assume that production is always automated only because technology allows it. Wrong, automation is function of the demand. You put this false assumption in the title of the post: technology does not produce unemployment. Automation produce reallocation of human resources and eventually unemployment. 
