Thanks for following up. You're asking about a number of issues at once, so let's see if we can distill them out.
First, you wonder why people must have "essences" at all. That's a big question -- Hume is a well-known philosopher who can be read at suggesting that persons or selves don't exist. That 'no self'' position is a minority view within philosophy, but is arguably the position espoused by Buddha and has some affinities with the 'eliminative materialism' defended by Paul and Patricia Churchland. I'd encourage you to explore those views further.
Second, in my earlier response, the point that we might identify someone on the basis of social facts about him or her was not an argument for genetic facts being essential to a person. Rather, my purpose in making that point is to illustrate how this line of reasoning is invalid:
Whether a is F can be reliably determined by whether a is G. Therefore, G is a's essence.
To see why, suppose (again) that the way we would normally identify Mandela is by reference to social facts about him: his personal history, accomplishments, etc. It wouldn't follow that those social facts constitute Mandela's identity. After all, there are other facts about a person by which we might be able to reliably identify him or her without those facts being essential to that person. I can pretty reliably identify the comedian Carrot Top by his massive orange hair -- but his massive orange hair clearly isn't essential to him. If he cut or colored it, he wouldn't be a numerically different person. So the general point is that even if some fact can be relied upon to identify something, that fact need not be that thing's essence. And that goes for genetic facts too: If I could reliably pick out a person on the basis of her genotype, it wouldn't follow that her genotype is her essence. Of course, if some property is a person's essence then of course we can identify that person by picking out whomever has that property -- but the contrary doesn't follow. So I was not arguing from the claim that we could pick someone out on the basis of her genetic constitution to the claim that her genetic constitution is her essence. That would've been a bad argument!
That said, you're correct that I didn't offer earlier an argument in favor of our genotypes being our essences. And in general, the philosophical dialogue around this issue tends to employ intuitions about what sorts of changes a person can undergo while still remaining that same person over time. As I said, I share some of your skepticism regarding genetic facts being our essence. But for defenders of that view, it seems intuitive to think that while virtually every other fact about a person might change (including social facts) without a numerically distinct person coming into existence, a change in that person's genetic constitution would result in a new, distinct person. If you find that thought intuitive, then you're likely to think that genetic facts are our essence. If not, then it would be worth exploring what's mistaken about that thought.