It's as close as you'll get to a random controlled experiment on this topic, because an actual controlled experiment would not be ethical. Studying people from the same school who lived in the same neighborhood at the same time controls a hell of a lot of the usual confounders.

True. Perhaps one possible confounder though is that parents who are attentive enough to think about lead paint and test for it / paint over it / remove it are also more clued in on other matters.

One of the studies I'm thinking of relies on the boundary effect, where children who tested with 100µg/dl of lead in their blood were all treated and those with 99.99µg/dl were not, and this threshold (I'm not sure exactly what it was) is way too high.

Right. The same issue confounds studies that show charter schools perform better than their public school counterparts, after controlling for student demographics. It could be that the quality of a charter school education is truly better, but it’s also possible that parents who care enough about their kids’ education to take initiative to send them to a charter school provide a better home environment in general.