> You're just as entitled to kill me as I am to kill you

This is exactly it. Your freedom is a function of your relationship to those who can use legitimate force against you - of how much power is held over you. If you can be killed without consequence then you're only as free as you are strong. This is a description of liberal anarchy, which is the natural state.

I guess I should have elaborated that the it is a liberal order which guarantees everyone an equal right do everything which injures no one else. It is the pursuit of order while maximising the freedom of the natural state that motivates the liberal. So the response isn't "killing like that is simply wrong", it's that violence without due process is disorderly.

> the position you're evincing here is liberal but anti-utilitarian

It's not actually, it's anti-authoritarian, which is a synonym for liberal. I'm arguing that the only legitimate use of (physical) authority is in the maintenance of a liberal ecosystem. It is not legitimate to use authority to intervene in the outcomes that ecosystem produces. You've not directly answered my question:

> What gives you or anyone else the moral right to intervene in someone else's family, presumably by force, over their objections?

The answer, should you produce it, would presumably justify any authoritarian intervention in pursuit of a utilitarian objective.

> prosperity is eminently attainable without liberalism

This was tried and failed in the Soviet Union for a reason. It will fail the same way every time power is concentrated in human hands. Liberalism-authoritarianism is a one-dimensional axis. Power is either diluted or concentrated. Neither outcome is utopian, but the failure modes differ. Giving relatively large amounts of power to the average individual produces all sorts of negative outcomes (eg. school shootings), but the consequences of concentrated authority are always catastrophic. The old adage about eggs and baskets applies. Distributed power is antifragile.

> Deciding to privilege non-intervention over any other course of action is itself a choice that can cause harm.

This is an opinion. I disagree. The harm is caused by the agent (person) or natural circumstance that triggered the outcome. If you get sick it's the disease that causes harm, not the person who didn't care to help you. If you're pushed out a window, it's the person who pushed you that caused the harm, not the one who didn't catch you.

> It injures the children.

This is also an opinion. People disagree axiomatically about what sort of upbringing constitutes injuring a child. Some would say failing to enforce attendance of religious school is injurious. You're sure you know best?

> Natural rights are a very flexible concept.

They're not really. It's pretty simple, you have the same rights you would in a state of nature with no other person intervening. So yes clean air and water, but no not the professional services of other people. Yes rights are only meaningful when they intersect with the rights of others. The entire liberal thesis is that the right of people to choose how to live supersedes the authoritarian pursuit of collective outcomes. We can still pursue them, just on a voluntary, consensual basis.