Can You Save Egalitarianism By Making It "Analytical"?

In each case, analytical egalitarianism reduces the risk of one set of bad policies, and increases the risk of another important set of bad policies. We can imagine cases where bending the truth would benefit the world, but this isn't one of them.

Let me be succint.

Societies that are not built (perhaps correctly) on the assumption of egalitarianism: Just about every society since time began except for some late Western societies.

Societies that are built (perhaps incorrectly) on the assumption of egalitarianism: The United States.

Both have flaws, but I know which one *I'd* pick.

Do you?

Might I suggest reading "How the Laws of Physics Lie" by Nancy Cartwright? Virtually every science is based on things that are known to be false. (Have you ever come across a frictionless plane? Neither have I) That the basics are false isn't what is at issue. It is whether they provide close enough approximations to the truth such that we can learn more from using these assumptions than we could otherwise. No model is going to be perfect, and most do not aim for realism in the strong sense that you seem to endorse.
In the case of egalitarianism, there are two programs one can consider. First, there is the traditional account that is trivially justifiable. For this, look to Book I of Hobbes' Leviathan. People are all considered equal insofar as no one is so much stronger than others than they couldn't kill him if they wanted to. No one is so much smarter than others than they couldn't band together to outdo him as well. This is a premise that is going to be hard not to accept. Of course, if you want to caricature the position by claiming that it states that everyone is precisely equal, then sure, it is easily refuted. But the core notion is merely that everyone has roughly the same abilities, whether they be physical or mental; no matter how powerful you are, a group of people can still knock you down.
That is the descriptive claim that is pretty hard to argue against. It has been a standard in political theory since at least Hobbes. One can try pushing the descriptive claim further, and argue that people are very nearly equal by nature, but again, one could simply view this as a modeling assumption, justifiable by reference to the weaker claim. One can make a further normative claim that people ought to be treated as if they are equals. This is where substantive moral and political theory comes from. A luck egalitarian could happily agree with you that there are some innate differences, and then respond: why should that matter when forming social policy? Those people that start out ahead didn't do anything to earn it - they just ended up that way. They won the lottery of abilities. But that scarcely entitles them to the full range of gains from that lottery. If we are behind a veil of ignorance, surely we would agree to spread the benefits of this lottery around. So why not set up our institutions in such a way to both promote growth by rewarding the utilization of those abilities, and promote justice by ensuring that that growth is used to help equalize the difference in innate capabilities?

There is plenty that one could latch onto for critique, but what you try to use is almost certainly not it.