Some might still call that a free market.
However lots of people misunderstand that theory and think the wealthy influencing the state is what libertarians want. I’ve not found any libertarian theoretician who advocates for that.
It's trivial to forget that their isn't much in the way of ground truth underneath society what people think of as the government not meddling much with society actually usually means leaving alone the current result of centuries of meddling if if the current distribution of tokens and things were the result of natural law and any change or redistribution is the only coercive act. If you imagine a board with certain positions privileged above others and spaces pre parceled out to certain players all the existing restrictions on players movements and actions are all coercive its just coercive in the ways that is presently expected as normal.
But I disagree that getting the government out of the way means leaving existing power structures intact. What I have learned from libertarian theory is that a whole lot of existing power structures exist (in the USA where I live) specifically because someone bribed the government in to supporting something they wanted. Take Disney continually extending copyright laws for example. Instead of getting old media in the public domain we get ever more restrictive copyright law because some corporations found it worth fighting for. Removing government in that instance would reduce existing power imbalances. So called “right to repair” laws for example provide a balance to the powers granted by government to corporations that would lock customers out of repairing their own equipment. I’ve also heard a few credible stories of instances where big corporations advocate for regulation that ends up squashing competition. Of course some kind of regulation whether from the state or by other means is necessary to avoid pollution, waste, abuse, etc.
In essence I am saying that even though there are many who call themselves libertarians who simply don’t understand the theory or advocate for its actual practice, this does not mean that libertarian theory is itself bad. I think instead that certain aspects of libertarian theory appeal to people who maybe just don’t want to pay taxes but never really learn or care about the theory behind the idea.
I've come to see "free markets" as the economic fabric on top of which we can build structures to support one another. For example we could form large healthcare collectives in a free market, as an alternative to state controlled health care (or insurance companies). I think a big factor is collective control versus minority control of the institutions that provide services like that.
As far as what a person without housing will do with the money - I'd rather offer the help than assume they don't know whats best for them. Even if they use it for drugs or alcohol as is often the source of much hand-wringing - then maybe it will help them get through the day. Of course, giving money to shelters is a good use of money too. And food banks.