Not true. A state could always justly protect people willing to be protected from initiation of force.

Fair enough.

But it cuts against the grain of the fundamental premises of both ideologies. That's what I'm saying. The zero order application of both ideologies leads to open borders.

Yes, I agree that open boarders is the most obvious policy for a libertarian utilitarian, although more from the libertarian POV. Utilitarianism requires thinking long-term - redistribution of wealth by tax is clearly a utilitarian good in the short term, but in the longer term it might become harder to incentivise useful work and so the issue is not so clear cut. This is why its possible to be libertarian utilitarian, rather than a communist utilitarian.

In the same manner, the long-term effects of absolute open boarders might be quite negative.