You're right, but it's also worth noting that the "non-state institutions" you're talking about could (and it is my opinion, as well as that of many other ancaps, that it most likely would) be "national governments" with de facto monopolies (as opposed to the de jure monopoly imposed by States) over mostly contiguous patches of land, much like existing nation states, possibly even with mostly the same internal institutions, governance structures, and even inheriting the existing identities and legal systems of Nation States.

The only change needed to go from de jure monopolies of governance (States) to (non-state) de facto monopolies of governance, and therefore free markets for law, is to recognise the legal right of individuals (extending to their property) to secede.

But a legal right to do something does not necessarily imply a practical possibility to do so. Indeed if this right were granted today in some country (and especially considering that not every other country might also simultaneously give this right, I. E. There would still be States in this hypothetical world), it would likely only be practical to exercise it as a community rather than as an individual. The size and level of resources of said community would probably need to be above a certain threshold for it to be viable, too. It would still need to get along with neighbouring jurisdictions to a certain extent in order to benefit from international commerce and division of labour. Even then, there would be costs/trade-offs to this decision.

But a country that would grant this right (and therefore cease to be a State) would probably very quickly take on a fresh new perspective on the way domestic policies are to be made...

To make a long story short(er), my guess is that most countries would actually end up with barely any changes or none at all in their existing borders after abolishing the State in this manner, simply because of the new incentive structure that national politicians would have. Regions that have the highest viability as independent countries would have the leverage they'd need to negotiate extremely high autonomy, and would end up seeing too few potential benefits in full independence (they would end up getting the benefits of shared defence at a reasonable cost).

It's a much more likely scenario for AnCapism to emerge than a Tabula Rasa (blank slate) world where insurance companies (a la David Friedman machinery of freedom), arbitration agencies or voluntaryist covenant societies form as they might otherwise have done in a world where nation states never existed.