Both the lead and the abortion hypotheses could be true. In fact they could both be true AND the papers indicating one explanation could conceivably throw out the other explanation. If you're interested in one hypothesis, you might not spend the time going through alternatives as thoroughly as someone investigating the other.
Looking specifically, the studies are always trying to isolate some effect, often using some sort of clever control derived from the proposed mechanism. So in this case there's a paper that points out how lead only goes into the water if the water is acidic, and they look at places with non-acidic water or non-lead pipes to control. For abortion, it's been a while since I read it, but there would be something along the lines of when abortion became widespread in a given community. You'd then look at differences in frequency and think about whether that came out in the crime stats when the kids got older.
The rabbit hole is where you can come up with ever more interesting corollaries and data that support or do not support some hypothesis. You see this across the social sciences all the time. There's also confounding problems, say you know people tend to move to the city when they grow up (I don't know if this is true), then how valid is your birth -> crime data?
Here's a load of random observations I thought up:
- If abortion is the key, why do crime rates keep falling? Why is it not a step effect?
- If it's lead, did crime rates increase 20 years after leaded fuel became a thing?
- What if it's abortion AND lead? There are countries where abortion and lead are timed differently, how about looking at them?
- What if it's neither, but both are caused by some other thing? Say economic development -> both?