I'm going to try to plough a middle furrow here. There are people who are very into the idea of indigo children and there are others who regard them as mumbo-jumbo. I don't really fall into either camp, as usual.
Science fiction, although generally about the present rather than the future, is often quite good at prediction, often with technology and its social consequences. I often feel that it's slightly spoilt by a tendency to change only one variable, like a scientific experiment. Just plucking a random example out of the air, Asimov's robot stories tend to be very much set in the mid-twentieth century when they were written with the difference that there are robots with similar intellectual capabilities to human adults. Change in the real world occurs along many lines at once, so for example we have the influence of social media following on from the advent of the Web at the same time as medical innovations changing the way we look at ourselves, genetic modification, fracking, climate change and a resurgence in religious fundamentalism combined with increasing tolerance of sexual minorities, just to pluck a few things out of the air. Then again, sometimes there are apparently single changes so momentous that history gets divided into before and after. 9/11 would clearly be an example of this, as would Tim Berners-Lee's invention of the World Wide Web.
One example of such a change depicted in science fiction, but somewhat tangential to the mainstream view of science and technology, is that of special children appearing with super powers, and I've just discussed this elsewhere. It's found in various other places as well, not always in such a positive light.
Jerome Bixby's It's a Good Life, published in the same year as Clarke's Childhood's End is particularly haunting in this respect. There's a similar character in Stapledon's Odd John, but the majority of superhumans in that book are basically good, though morally ambiguous from a baseline human perspective. This moral ambiguity reflects alternative ways of seeing children.
A somewhat similar idea applied to real life is that of Indigo Children. The name comes from Nancy Ann Tappe about forty years ago, who said that she saw indigo auras around some children whom she expected to be special in the new age to come. Indigo children are said to be highly intelligent, not compliant with schooling, empathic, strong-willed, having strong innate spirituality, intuition and purposefulness. They are also seen as strange by others and are described as having a strong sense of entitlement. I have come across children described as Indigo myself, as it was a popular in the 1990s and 2000s in home ed circles, which considering their perceived resistance to schooling is unsurprising.
It's difficult to mention the idea of a sense of entitlement without seeing it as a criticism, but in this case it seems to be seen as positive. I don't know what I think about that to be honest, but then again, I don't know what to think about the whole thing, However, there are other people who do know what they think about the whole thing, and it's not the same as what the indigo children's parents think at all. Some psychologists take the view that the construction of the concept of these children is a response to the diagnosis of particular developmental issues in them such as ADHD, ADD and autism. The idea is that parents of indigo children prefer to see them in this way rather than as labelled as in some way defective. That idea suggests a certain degree of self-assurance on the part of both the parents and the healthcare professionals involved, and also an idea that the current social status quo is in some way the right way to be, or in some way unchangeable. It is possible that this is the case, of course, but in view of the fact that this society gives the impression of always having been massively dysfunctional, I don't see it this way. My view of diagnoses such as ADHD and autism is that where they are accurate in terms of criteria, they reflect a poor fit between the cultural milieu and the way the person receiving the diagnosis is, which may or may not be helpful for them. It may be helpful as a source of explanation to them for their difficulties which suggests a way forward, and there's no real need to explain why it might not be helpful.
There is a more significant problem with the idea of indigo children, namely that it sets certain children aside as more special than others. Many people whose children's education doesn't include time at school would say that schooling simply fails most children both as a means of letting them be children and as a way of enabling them to achieve anything like their full potential. Hence the criterion above of children who are not compliant with schooling could apply to almost any child in many people's views. That said, there is a growing issue of various degrees of non-neurotypicality among children recently, in areas such as being on the autistic spectrum, dyslexia, ADD and ADHD. I would also suggest that gender non-conformity and pathological demand also belong here. The question of why this issue is growing could reflect increasing recognition of the issues, medicalisation or increasing prevalence.
Just to depart into a somewhat fantastic realm for a while, just suppose that something like the scenario described in science fiction of really "special" children did emerge. Suppose, for instance, that children clearly began to demonstrate abilities such as telepathy, telekinesis and precognition as a matter of course. How would the system deal with them? Would it be able to accommodate or nurture such abilities? Would it even recognise them? Or, would children with telepathy and precognition be considered psychotic and medicated in such a way that they ceased to have such abilities? Or, would it be a case of such children being sidelined and just not being accommodated in anything which would enable them to develop their talents?
Getting back to the real world, these stories and the idea of indigo children, even if they reflect nothing else, communicate the truth that children, in the form of the next generation, will eventually take over the world, and when they do so, they will need to have the adaptability and resourcefulness to find solutions to the problems previous generations have presented them with. In order for that to happen, they need to be able to develop whatever talents they have and engage with such talents with the world. If schooling can't find a place for children to do this, it needs to be replaced, because now a utopia has become a pressing need, not something we can even survive as a species without.