Here's how I think about our methodology. Car seat laws generate an unusually sharp prediction for births that can be distinguished from a lot of competing effects (not perfectly, of course, but compared with a lot of other demographic effects)
We're not testing whether the state and years with more stringent car seat laws have lower birth rates. That could be due to tons of stuff going on in those states.
We're not even testing whether the states and years with more stringent car seat laws have lower birth rates just for third child births. That could be relatively fewer things, but still a lot.
Rather, we're testing whether states and years with more stringent car seat laws have lower third child birth rates specifically for families whose first two children are below the mandate for their state and year. And this mandate is itself changing over time for both states and years, so we're talking about different ages of children in question depending on which state and year it is.
This clearly doesn't rule out everything. But the set of candidate explanations for what else could be going on for those specific families in those years is a lot smaller than the broad demographic effects we started with.