Unjust inequalities are of course undesirable, but not every inequality is unjust, and excessive equality has its downsides as well. Not all people are the same, and the differences between people make some of them better than others. Trying to ignore that fact results in a society that ends up discouraging people from fulfilling their potential. There is an analogy here to the artificial economic equality enforced under communism. Artificial economic equality treats everyone as though they were equally productive. And this has the effect of discouraging anyone from being productive, since there is no incentive. Additionally, by failing to reward productivity, communism also has the effect of leading people to forget that productivity even matters. Similarly artificially treating everyone as equal discourages people from being more than average, and by not celebrating greatness we are led to forget that it even matters.
I think there are primarily two reasons why modern societies seem intent on perpetuating the myth that everyone is equal (at least in intrinsic value if not in purchasing power). The first is that recently most societies have undergone a process of self-improvement in which a number of unjust inequalities have been eliminated. Because of the difficulty in eradicating these unjust inequalities we are left with the suspicion that any inequality is an unjust inequality. I grant that it is appropriate to be suspicious of inequalities between people, because they can do a great deal of harm. However, it is relatively easy to separate most unjust inequalities from justified ones. An unjust inequality differentiates people based on what they are, while a just (or at least not unjust) inequality is based on who people are. Differentiating people based on what they are means differentiating them based on some external property, such as who their parents were, which is of course the source of most inequalities since that is the easiest way to separate people into different groups. In contrast differentiating people based on who they are means differentiating them based on internal properties, properties of their personality or character (of course we only have access to those properties by observing external properties, like their behavior, but all this means is that our judgments are fallible).
The second reason that the myth of equality is perpetuated is that it is flattering to the ego. It is nice to believe that there is no one who is better than us, because it makes us feel big. Even when someone is shown to be better than us at some specific task we can then comfort ourselves by appealing to the idea that we are equal, and thus that there must be some way in which we are better than them. Of course this is fallacious reasoning, but it is psychologically appealing nonetheless. Now I’m not trying to accuse people of purposefully promoting certain ideas about equality just to make themselves feel better, but I am claiming that when those ideas are introduced, for whatever reason, this fact biases many people unconsciously in their favor, making such equality seem better than it really is.
As I already mentioned one effect of this artificial equality is that people are discouraged from striving to be better (or even from realizing that they should strive to be better), and thus we end up with a society of mostly average and unremarkable people. Depending on your attitude towards humanity that may or may not be a bad thing. If you are of a more classical persuasion you might feel that personal excellence and nobility are the whole point, and thus that a society that doesn’t encourage those characteristics is a disaster. But if you are of a more modern bent then you may think that all that matters is pleasure, in which case individual achievement doesn’t matter. But whether people are led to excel or not is not the only consequence of treating everyone as equal. Another effect is that it makes people a poor judge of others, which can be disastrous if your political system involves electing officials. If everyone is equal then who you prefer is basically a matter of personal preference, and you are free to make your decision based on whatever criteria you like. Which means that when it comes to electing officials someone who is objectively better (by virtue of being more competent) may be passed over for someone who is more likable. Of course just permitting ourselves to consider some people as better than others wouldn’t automatically make people better voters, but it would allow candidates to actually compare their personal strengths to each other (in addition to the issues) without sounding pretentious (as it stands now no candidate can directly claim that they make better decisions, are more responsible, or are more intelligent without falling out of voter favor by violating the assumption that everyone is equal).