I don't think that "everyone is inherently equal" is a conclusion you can reach through logic. I'd argue that it's more like an axiom, something you have to accept as true in order to build a foundation of a moral system.

This may seem like an arbitrary distinction, but I think it's important to distinguish because some people don't accept the axiom that "everyone is inherently equal". Some people are simply stronger (or smarter/more "fit") than others, they'll argue, and it's unjust to impose arbitrary systems of "fairness" onto them.

In fact, they may believe that it is better for humanity as a whole for those who are stronger/smarter/more fit to have positions of power over those who are not, and believe that efforts for "equality" are actually upsetting the natural way of things and thus making humanity worse off.

People who have this way of thinking largely cannot be convinced to change through pure logical argument (just as a leftist is unlikely to be swayed by the logic of a social darwinist) because their fundamental core beliefs are different, the axioms all of their logic is built on top of.

And it's worth noting that while this system of morality is repugnant, it doesn't inherently result in everyone killing each other like you claim. Even if you're completely amoral, you won't kill your neighbor because then the police will arrest you and put you on trial. Fascist governments also tend to have more punitive justice systems, to further discourage such behavior. And on the governmental side, they want to discourage random killing because they want their populace to be productive, not killing their own.