The point about competition to control the central sub-empire is an interesting one. If I am assigning the terms correctly, in the Musk empire Musk himself serves that role. Is that correct? If so, then I wonder if that helps to explain the prevalence of a single dominant player in wildly successful empires - you can't effectively compete for control over one person, so this acts as a constraint on the amount of resources wasted in that way.

I think you're eliding an important distinction between empires and kingdoms in your initial definition.

A kingdom is a patriarchal polity governed by a single person as though it were his household. Setting aside fractal kingdoms (e.g. European Feudalism), a kingdom is typically a single, unified system of survival. Insofar as this household is considered legitimate by its members, they are comparatively tractable when asked to behave in ways consonant with that framework, but hard to push outside their role, irrespective of self-interest.

By contrast, an empire is more typically a polity (of any kind) that conquers and rules other polities, without integrating them into itself, so that they retain a sort of distinct identity as potentially autonomous provinces.

When the central polity of an empire is an individual person, you have what the Greeks would call a tyranny, but there are many other ways to arrange empires, including ones in which social expectations bind the central institution more, but are typically more stable.