Self reported is unfortunately very common in nutritional studies, for obvious reasons. I don't think it makes sense to dismiss a single study for having the same flaws as most others, as opposed to the general recognition that nutritional science findings are baseline less robust.

I wouldn't cut them so much slack. The study is either reliable or it isn't. And it wouldn't take them much more effort to make the results significant in some way.

In this case, I think that just giving these children the self-reported diet would yield massive effects on their health, sugar or no sugar. I am quite sure children with obesity problems don't know their real diet - not because they are dishonest, but because we (as people) are very efficient at hiding the uncomfortable truths from ourselves.