I think the behaviour Nigel describes in the OP (and yes, I think it is a very good question) is partly a spandrel of social evolution. I also think it is partly also caused and sustained by the prevailence of neurosis among people raised and living in the multicultural (and, as someone pointed out earlier, though I don't remember who and which thread, fractally cultural) environment of the megapopulation after being raised in more or less dysfunctional families. I believe this unfortunate story began about the time the paelolithic slowly turned into the neolithic, whenever that was. When sustained and significant interaction between cultures and hence cultural evolution got off the ground.
In greater detail: I think people do not find it hard to admit to making a mistake in general. I think they find it hard to admit a mistake they are confronted with when they feel misunderstood, and when they sense that they are being rejected. Of course, some people are maladaptive to the point that they feel they are misunderstood and rejected every single time they are confronted with a mistake (this is the sort of thing you are trying to pin on me, totally unfairly, but I'll leave that until later.) It is those element of misunderstanding and rejection that make it hard. And it makes it hard because being misunderstood and rejected is actually quite terrifying.
And this is despite the fact that, if you look at it objectively, in this day and age, being misunderstood and rejected is in most cases not such a big deal.
But it is a big deal in a monocultural tribal society. In a society where there is one language, one ethos, one set of customs, one way of understanding the world, in which people who stray from those norms are seen as fundamentally defective (mad, or evil, possessed, or whatever, but badwrong), being misunderstood is a terrible thing. A terrifying thing. It could well be the lead-up to being abandoned, or coerced. And most people (those whose early years are spent in a community that is functional enough) spend those first three, most formative years of their lives, when their emotional self-regulation is fine-tuned for a life in a particular culture, in an environment (a family, or, unfortunately, sometimes an insitution) which is quite a lot like a monocultural tribal society. So their emotional self-regulation, when they are thrown into the sea of the megapopulation at age 3, or later (kindergarten, school, etc.), is that of a monocultural human. Very scared of being misunderstood/rejected. Try to think back to your earliest memories: your were a blessedly happy and sheltered child indeed if you don't recall some scary incidents that involved interacting with strangers who did not know how you tick and didn't much like you.
If close family did that, so much the worse, which brings me to the second part of my explanation:
Most of the people on this board, just like most of the people who grew up in this civilization we share and are alive today, actually were raised in a manner that was far from optimal (in the evolutionary sense), and hence their emotional self-regulation is (to a greater or lessed degree) off-kilter. They feel threatened when they are faced with their mistakes, because they think they can only be loved if they are perfect: their lack of security in their relationship with their primary caregiver scars them for life. Some overcome it. Many never do. Those who don't often find it very hard to admit being wrong because they are afraid that if they do, they will be left alone to die. Those two effects interplay and reinforce each other in a number of interesting ways.
Since you don't seem to understand evolutionary biology, let's improve your argument.
A spandrel, sensu Gould and Lewontin, is a character of a species or higher group of organisms that does not have adaptive significance. It is a product of evolutionary contingency, but does not serve any adaptive function. As an aside, he tendency to assign adaptive value to every character is known as the Panglossian Paradigm, also in that same paper.
So, what you are saying is that the unwillingness to admit mistakes has no adaptive significance, but the measure of adaptive significance is fecundity, not emotional well being. Your argument should be that the unwillingness to admit mistake has no significance on fecundity. Furthermore, your argument revolves around a behavior that has lost adaptive significance recently, which is not the character of a spandrel. Instead, in your argument, the behavior should be called vestigial.
But even with these changes your argument is unfounded. There is no evidence that an unwillingness to admit mistakes has loss adaptive significance, i.e. has lost its impact on fecundity, the /only/ relevant measure of evolutionary success. Nor that it is socially unacceptable (any gander at political battles will show this). Furthermore, there is no "optimal in the evolutionary sense", only "whatever works".
*goes back to work on actual biology*