is correlated with [...] which is correlated with [...] which is genetically inherited from [...] which is correlated with

I agree with your conclusion that the heritability of firstbornness is nonzero, but I'm not sure this reasoning is valid. (Pearson) correlation is not, in general, transitive: if X is correlated with Y and Y is correlated with Z, it does not necessarily follow that X is correlated with Z unless the squares of the correlation coefficients between X and Y and between Y and Z sum to more than one.

Actually calculating the heritability of firstbornness turns out to be a nontrivial math problem. For example, while it is obvious that having few siblings is correlated with being firstborn, it's not obvious to me exactly what that correlation coefficient should be, nor how to calculate it from first principles. When I don't know how to solve a problem from first principles, my first instinct is to simulate it, so I wrote a short script to calculate the Pearson correlation between number of siblings and not-being-a-firstborn for a population where family size is uniformly distributed on the integers from 1 to n. It turns out that the correlation decreases as n gets larger (from [edited:] ~0.5[8] for n=[2] to ~0.3[1] for n=50), which fact probably has an obvious-in-retrospect intuitive explanation which I am somehow having trouble articulating explicitly ...

Ultimately, however, other priorities prevent me from continuing this line of inquiry at the present moment.