US libertarian party is based on small government but it very much wants a strong military. Some members are anarchists but the party is far more diverse than just that. So, referring to libertarianism in the US as identical to anarcho-capitalism isn’t accurate.
Anyway, I agree that Rothbard doesn’t want government, but I am saying he does want the rule of law which is a separation point from pure anarchism. How exactly you get a court system that’s not backed up by the threat of force without people simply ignoring it is a point of contention.
Anyway, Rothbard, a prominent anarcho-capitalist, states quite clearly that "libertarians" (i.e. anarcho-capitalists) are opposed to government, which contradicts your statement that anarcho-capitalists want "a strong government".