Oh Grow Up!
The stereotype is wrong
It's based on too little evidence and poor interpretation. Ariès relied heavily on pictorial evidence, on works of art. This is tricky evidence that cannot be read at face value. Secondarily he relied on didactic literature, which by definition has a particular point (often moral) to make, and again needs to be read and interpreted carefully.
More recent historians have looked at other sorts of records, such as court records, wills, documents associated with orphanages, and Church law. Each of these emphasizes a different aspect of life. It's a mistake to generalize from any one source without examining the others as well. These historians paint a much fuller, and more sensible, picture of the stages of life.
If it's wrong, why is it widespread?
Because no one listens to historians.
The myth is widespread partly because it gets repeated endlessly in various forms and media, but that really begs the question. Fundamentally the myth is perpetuated, despite the evidence, because it reinforces our ideas about ourselves and about the past. This topic really deserves a separate essay, but I'll state it briefly here.
Modern people (by "modern" here I mean after industrialization) need to believe that things are better now than they used to be. For a very long time, the Middle Ages has served the role as being "worse" so that we can be "better" (I have to note that the Middle Ages has a different role for the Romantics, but that's a different topic). Even where a particular myth doesn't feed directly into this need, the idea that the medieval past was backward, primitive, and brutal provides the underpinning, the necessary foundation on which our ready acceptance of the myth rests.
So, between the constant reiteration afforded by mass media, and a predisposition to accept certain kinds of statements, we have ideas like these about childhood persisting. It doesn't matter what the historians say, because the historians don't speak to a social need. The myth does.
One final point: notice how many myths are about the Middle Ages, either explicitly or implicitly. They don't apply to tribal societies, to China or India, nor even to ancient Greece or Egypt. This is particularly true of myths relating to society and social relations. This is another bit of evidence that there's more going on here than simply misunderstanding the past. It's a particular kind of misunderstanding of a specific past. That's why I think "myth" is the appropriate word to use.