Edit: Re-reading the question I do think this somewhat misses the point. So I wanted to add something about making it more believable directly.
One big thing I think would be historical depth. Don't just tell me about your free market version of Yelp - I want the story about how it came to be, and how you kept it on the straight and narrow, and the computer app that aggregates free-market inspections and makes it simple and easy to use. Tell me about competing companies of inspectors, and certification companies that watch the inspectors (and each other) going back hundreds of years. Don't just tell me about EvilCorp polluting a river and killing hundreds of people, give me a story about how the city banded together, proved that it was them and took justice, and set up a schedule to keep it from happening again. Give me the bad side, the gritty details. There's an applicable quote to this effect:
Laws are like sausages, it is better not to see them being made. - Otto von Bismarck
You need to show me the sausage - all the bad stuff, and how systems were built to respond to it. And how those systems then respond to adversity, and Bad Actors (I like that phrase).
Hope this helps.
I'm someone who reads stories like that and I think "this is a wishful fantasy" - most of them seem to have the same naivete as young adult fantasy (good guys always win, bad guys will eventually be punished, most people are reasonable and good at heart if the government would just get out of the way, etc).
These are the things you'd need to address to make me take your story seriously. I'm not saying that it's impossible to do so - just that you need to show me some sort of realistic mechanism that fits your philosophy. Or, I guess, you could just accept that bad things happen and live with it.
1. Slavery / Abuse.
As you mentioned, this is never shown. But I don't understand how it could be prevented. The usual answer in these books boil down to "good thinking people will step in and stop it" - usually through some mechanism like a self-organized militia, or calling out offenders in duels. But does that mean if someone comes in with bigger/better guns, they get to do what they want? Can a really good duelist basically go around raping, murdering and stealing with impunity, because he kills anyone who calls him out? It always feels unrealistic.
Another way this shows up is communities with differing values. I'll read a story about a free market/anarchy city, and it all works well because everyone is pretty much from the same culture. But I wonder what would happen if an extremist religious group set up shop next door, claimed the same rules and then started marrying 12-year old girls off to old men under the umbrella of religious freedom? How do you stop them from declaring that the girls are consenting adults, and no they don't feel like talking to you because they're busy in the kitchen? Try back later.
2. Consequences of zero regulation.
I'm more than willing to admit that, say, the current US government over-regulates and that causes harm. But no regulations at all seems even worse. If you have a perfect information society, it might work. Bob creates a restaurant and sells you bad chicken, someone eats it and gets sick. Everyone realizes what's going on and stops eating at Bob's, and he goes out of business. It's a nice scenario.
But what about in a non-perfect information system? What happens when Bob starts a misinformation campaign, and blames his supplier? If when he realizes what's happening he closes up shop and moves to the next city down the road, and starts his business up again where no one knows him and does it again?
3. Negative Externialities
This is an economic concept where the cost of an activity is, at least to some extent, taken up by people who don't benefit. And it's one of the big areas I believe in regulations for. Consider a business that's using an extremely polluting technique to create cheaper goods. This pollution is non-obvious (toxic dust or gases in the air, for example), but will have long-term negative health effects for everyone in the area. How do you address this without some sort of government body to do inspections?
You could have watch dog groups, but would they really be funded? If so, how do you manage it without basically re-creating a government? How do you prevent the corporation from taking Bob's strategy and using misinformation to counter that, or fake watch dog groups that give them a clean bill of health? How can you expect a random citizen to be able to sort truth from falsehood in this kind of scenario?
These are, FYI, precisely the kind of solutions that I find to be incomplete. They work on the surface level but as soon as you consider them in depth, or consider what a criminal would do, they fall apart. They sound logical, but one of your assumptions is that it's a perfect information world - that everyone knows everything else - and that's simply not the case. Think about what happens when corporations, businesses and people lie and fake information. Your system has no way to resolve that type of conflict.
And since it needs to be clarified, yes - governments also have all these problems. They are not perfect at preventing violence, abuse, slavery, pollution, or food poisoning. But they do have systems to handle these problems. Are modern governments ideal, in any way? No. But they at least have some way to handle these issues. Anarchy/Free Market just seems like it has hope.
If there are no regulations, what do you do when McDonald's buys Yelp without telling anyone and starts controlling reviews - posting better results for themselves and sabotaging their competitors? How do you prevent smaller-scale false information from corrupting the system? If there's no central authority, how does someone looking for a bit to eat straighten everything out without doing a week of research?
2. Factory Inspectors:
Who pays for them? Do you go around and try to convince people that they're sick and collect funds? What do you do when the factory provides documentation from another inspector stating that they're safe, and how do you convince a court (and what kind of court do you even have) which one is valid and which isn't? Why does the factory even let your inspector in?
3. Social Contract re: Freedom
Have you defined a minimum age of consent? And what gives you the authority to go into someone else's house/community and determine that they're marrying girls against their will? Is hearsay enough? How do you prove that it's happening?