It does correlate, and it is accounted for. We have evidence to believe the causal arrow points lead=>poverty. This is because the natural experiments of phasing out leaded gasoline in different countries at effectively randomized times produce subsequent decreases in lead poisoning (and downstream crime). This is as close to a controlled "give one population lead poisoning" experiment as you can ethically run.
But there is also poverty => poor housing in less desirable areas => more pollution in general. Including Air pollution, car exhausts, lead from leaded gasoline.
This is well known e.g. (1) and seems like it would be a large effect.
I'm not saying that this invalidates the studies on lead, I'm sure that the researchers know how to do it far better than I do, and have spend a lot more time on accounting for it; just that poverty drives increased pollution exposure, as a general rule.
For poverty effects to dominate across countries, there would need to be a causal relationship between phasing out leaded petrol and countries' distribution of income; it looks implausible at first glance. It's much easier to phase out leaded petrol than to make persistent changes in the distribution of income. It would be very surprising if all countries not only did so, but did so at the same time.
Quite, phasing out leaded petrol at different times is the kind of "natural experiment" that can demonstrate the effects of lead reduction not poverty reduction.
This is not the same thing as "poverty drives increased pollution exposure, as a general rule" which is still true.
Other things drive pollution exposure too, like general use of leaded petrol.
