It’s important to note that Jefferson’s words (by way of George Mason, by way of Locke, etc.) are a statement of ought more than anything. The concept(s) of natural rights/natural law in Locke’s conception are not about our individual characteristics so much as moral truths, generally applicable, and how a government should execute upon these moral truths.
It is also important to note that there is a difference between inequality and differences. In a society it is how we - or more to the point, how a government - treats people that establishes their equality. The aim of Jefferson’s formulation is to assert natural, god-given rights. The existence of these rights is not dependent upon biological variation, and can exist in light of these variations. However, in practice, biological variations have obviously had implications for the rights ascribed to people.
After all, though, I’m having trouble understanding what you mean by "the only way to eliminate inequality is to treat people unequally." That is, all in all, what is the basis upon which equality and inequality are being evaluated?
I think that the main issue I have is with these three types of equality you’ve mentioned (I have not read Hayeck’s work, mind you). They appear to be quite gross in scope, and the way they interact appears to be quite restrictive.
Factual equality does not necessarily guarantee that we are unequal per se. That (2) is equal to (1 + 1) is necessary to the mathematical utility of those terms, the basis of evaluating their truthiness being mathematics as they exist in our reality, and the laws therein. Without context/a sufficient basis, a human cannot be evaluated in the same way. If you were to say that one man is born faster than another, and in this given context the ultimate aim of biology is survival, and speed ensures a greater survival rate, then yes, one man is unequal as compared to the other. But in the context of a society in general, and a government, utility is relative, and secondary to equality in terms of natural rights. In other words, biological differences, in theory, do not guarantee effective inequality.
Legal equality is easily understood if it’s meant to mean that we are all treated as equal before the law. But here, at least if you wish to invoke the ways in which this might extend to economics (or labor, or ownership, or taxes) you have to define equality further. Moreover, you must mind that laws are formed by people, in a process of consensus and consent. It is conceivable that legal equality, if law were formulated accordingly, could exist such that it does not clash with material equality. But I think this is an issue greater than legality, that is, one which thrusts us into questions of desert, how we ought to live in relation to others, how value and meaning are assigned to aspects of life, etc. - if we are in fact conceptualizing, reasoning, and not reacting to things just as they are. What’s more, if a progressive tax structure in and of itself signals a breach of legal equality, then again, this conception of equality is I think overly simplistic, too static, and - if not described in Heyeck’s book sufficiently - favors a labor theory which requires explanation.
If an example of material equality would be redistribution of wealth, and if the supposed, corresponding inequality would be that a tax law used to facilitate such a redistribution might include unequal tax rates (thereby, I presume, indicating some kind of legal inequality), then material equality as is is too simplistic. And to say that this form of equality is all socialists yearn for is I think a careless reading of socialism, or at best, sounds like a reaction to socialism akin to those found in social media more so than a rebuttal to the actual, relevant literature.
When all is said and done, what it seems like you're saying here is that if some form of equality is to the exclusion of some people's desires, it therefore makes it such that those people - if this aforementioned equality is pursued - must be treated as unequal. If this is what you're saying, then I think the conversation to have is, again, about how we ought to live, about what we perhaps should want, or expect, or how inclusive or perfect laws/society/a government can/should be, etc., etc., etc. But then I might have this all wrong and not know what you're saying at all, haha.
Anyway, upvotin’ ya. I like the fresh take, and I like the philosophically-geared posts :)