Probabilistically, it sounds like the study found P(hyper|dye) = P(hyper|~dye), that is they rejected P(hyper|dye) > P(hyper|~dye), and concluded P(hyper|dye) = P(hyper|~dye) (no connection) correctly.

You are making the same mistake by ignoring the quantification. The test used to reject P(hyper|dye) > P(hyper|~dye) uses a cutoff that is set from the sample size using the assumption that all the children have the identical response. They didn't find P(hyper|dye) = P(hyper|~dye), they rejected the hypothesis that for all children, P(hyper|dye) > P(hyper|~dye), and then inappropriately concluded that for all children, !P(hyper|dye) > P(hyper|~dye).