It's a valiant attempt, but I think this approach is fundamentally flawed, for three reasons.

First, are you sure you can define anything at all? You seem to be rejecting a biological definition on the basis of it being a bit tricky to classify every case. But there are always tricky cases. For instance, what is a knife? Can anyone really say? Maybe the definition of "knife" is up to the person who wants to bring their implement on the airplane?

Second, you are arguing that a society that sets out categories for different treatment should believe you about which category in which to put you. But why should *you* be the one to make the call? If it's not a clear definition, how can you possibly argue your case? Seems like society can just say, "No, we want you in that box," and you have no arguments with which to rebut them. Whether it's being a veteran, a student, a senior, a police officer, a diplomat, whatever--the distinction in category is almost always something that can be verified independently of the individual's claim.

Thirdly, your explanation fails to account for the fact that we observe roosters and hens and male lions and female lions and male and female fruit flies, and so on, and we don't generally have any trouble classifying at least the overwhelming majority of them. There seems to be some classifiable regularity, something called "sex". The animals don't self-report, and yet we seem to very reliably sort them. Biologists absolutely do this all the time. (Learning to sort male and female fruit flies is basically the first thing you learn how to do when you enter a fruit fly lab.) If we can do it for animals, and humans are animals, then maybe the supposed difficulties actually aren't that important, so we should avoid discarding the phenotypic and genetic approaches? (In fruit flies, you can do experiments on the sex determination pathway that results in flies with with atypical phenotypes--the solution there is to not call them either males or females outright, but subdivide by tissue type or whatever if appropriate ("somatic male, gonadal female" or "neuronal female, non-neuronal male").)

So I don't think this line of argumentation succeeds. This doesn't necessarily mean that your conclusion is wrong, but it doesn't seem that this is the reasoning one should use to arrive at it.

Now, you do appeal to what "actual people who study actual biology are fairly united on". But if your entire argument actually boils down to that, it's essential to cite them so that people evaluating your argument can go evaluate their arguments since you really just end up deferring to them.