Nice post. This is pretty close to the line I take when discussing libertarianism with skeptics, and it's one of the reasons we try to include optimistic articles about the state of the world over at Libertarianism.org. But it also risks being something of a straw man.

In a sense, it's similar to arguing that all you need to justify political libertarianism is the Non-Aggression Principle. There's an undefined term that's hiding much of the argument. For the NAP, it's what counts as aggression and as property. It's not that, say, a progressive who would reject strong property rights libertarianism thinks that it's okay to aggress against people or to take their property, it's that the progressive doesn't think taxation or regulation count as aggression.

So with your argument, "works" is doing a lot of, well, work here. What you're arguing above is that libertarianism as a political system defined by free markets and minimal or no government intervention and taxation "works" because in our daily lives treating each other with respect, not using violence, and refraining from stealing are the preferred behavior. Given that political libertarianism, then, is just taking those preferences to their logical conclusion, we in fact have considerable evidence that libertarianism works.

The counter is that libertarianism as a theory of government contains a good deal more than simply saying we should all respect each other's property, and that we don't have (as much) evidence for those parts of it. So your interlocutor isn't saying, "Libertarianism lacking evidence of its working because I think it's just fine to steal from people or punch them for no reason." He's saying, "Libertarianism lacks evidence of its working because, in order to have a peaceful and functioning society where we can engage in respectful, non-violent exchanges, we need a powerful government setting up rules, protecting our rights, enforcing agreements and laws, providing welfare so people aren't desperate, and taxing us in order to pay the costs of doing all that." That we all agree not to hit or steal from each other doesn't itself say much in response to that latter line of argument.

Which is all just a too long way of saying that, while it's important to point out that all of us to some extent internalize libertarianism principles, it's not quite enough to prove that, just because we live by those principles in our personal interactions, we'd benefit from those same principles applied at the state level if it means ultimately drastically shrinking or abolishing the state and its many programs.