I'm pretty confident philosophers do want to make sense of personal identity. But you are taking issue — not unreasonably, in my estimation — with a claim many philosophers make in motivating the non-identity problem.
Let's review the reasoning that's supposed to generate this problem. Let's imagine Nelson is brought into existence in circumstances C1. In order for Nelson to have been harmed by being brought into existence in C1, then there must be some other circumstance C2 in which Nelson could have been brought into existence which would have been better for Nelson. But (the reasoning goes) in any circumstance beside C1, the individual brought into existence would not have the same genetic constitution, and so would not have the same identity, as Nelson. Hence, there is no other circumstance into which Nelson could have been brought into existence, and so Nelson could not have been harmed by being brought into existence at C1. (And the point generalizes: No one is ever harmed — or benefitted — by being brought into existence.)
You are questioning the claim that had Nelson been born with a different genetic constitution, he could not have been Nelson. This claim can be expressed more precisely:
A person's genetic constitution is uniquely essential to her identity as a person.
I.e., all other facts about person can be different while she remains the same person, but a person's genetic constitution cannot. Let us call this claim G.
One point you make to criticize G can be dismissed pretty readily. Names are entirely contingent and so don't establish identity. If I had been named Django von Mozart, I would be the same person with a different name.
Your stronger point (suggested by your remarks about "purpose or context") is that identity is better explained by something other than genetic constitution: a person's accomplishments or other social facts. Let's us call your claim S.
Particular social facts about a person are uniquely essential to her identity as a person.
That's vague but good enough for our purposes. The question at hand: Why prefer your S to G? I have my doubts about G, but here's how I think defenders of G, and those who think the non-identity problem is a genuine philosophical puzzle, would argue for G over S.
First, it's important not to confuse the facts by which others know or identify a person and the facts that constitute his or her identity as a person. The way you and I think about Nelson is in terms of social facts about him (his accomplishments, etc.). If I were struggling to remember who was South Africa's first post-apartheid leader, you'd naturally tell me, "Don't you remember? It was Nelson Mandela." In other words, defenders of G may claim that your S does well in telling us how we pick out a person, but doesn't tell us what facts constitute their being that person. (This also bears on the point about names: Names are ways of picking out or designating persons, not constituting them.)
For defenders of G, the answers to your hypotheticals are clear: In the alternative reality in which someone conceived two months later than Mandela (or conceived from a different egg) was conceived in our reality did the same important things as Mandela did in our actual world, that person isn't Mandela. Mandela's genetic constitution is essential to him: He's not Mandela unless he has that genetic constitution. Everything else about him, including all of these social facts, is contingent. Suppose that Mandela had not become a student leader at Fort Hare; had not joined the ANC; had not been imprisoned at Robben Island; etc. That would entail, according to defenders of G, that Mandela's biography was different -- not that the person in question isn't Mandela. Your hypothetical Mandela, presumably, would just be an ordinary person.
G appeals, I think, to our sense that genetic facts make us up: most everything else about us can change but we remain essentially the same. I probably haven't said enough to convince you of G, but I don't think its defenders are just "stipulating" truths about identity.