The problem is the concept of "crime." It doesn't exist in a voluntary society. You can't commit a crime, you can only commit aggression. And the only question is what is appropriate restitution for that aggression. And then you go down the private courts rabbit hole.
So, would there be debtor's prisons in a free society? Would there be slavery? I doubt it; I believe the market would come up with better systems. A true free-market society itself would be a hell of an incentive to live productively. No one is collecting restitution from slaves or prisoners. And at the end of the day I think people want restitution much more than revenge.
You are absolutely correct that there is no "crime" without the state - there are only torts (actions brought by a victim for restitution and possibly punitive damages).
Though I think you underestimate the latent tendency for violent revenge that people harbor. Many people consider fist-fights for looking at their wife or daughter the wrong way. Many people would want to run your car off the road if you accidentally spilled coffee on them. Many people want to "turn the Middle East into glass" for what 19 Saudis did over a decade ago. Many people forget the New Testament "turn the other cheek" and angrily perpetuate the Old Testament's "eye for an eye".
The best way to handle this is, of course, through competitive courts issuing judgements open to appeal to other independent courts, not through courts beholden to the political pressures of mob-rule. But we can't fool ourselves into thinking that people aren't currently still in battle-mode (mostly due to the fact that the govt system is effectively saying "here's a paddle, you can go wherever you like, but we're all in one canoe, and we all have our own paddles.")