Is there a cryptographic reason for using an irreducible Goppa polynomial $g$ in the McEliece scheme? One doesn't need irreducibility to define a usable code, so I assume there is some structural attack against reducible polynomials? [One caveat is that the presentation I've seen for Patterson decoding uses irreducibility, but one doesn't need to use that algorithm (and it isn't used in e.g. the FPGA implementation here).]
The key generation is already annoying enough without enforcing irreducibility IMHO. The only thing I can think of is that irreducibility definitely ensures that the support $L$ is disjoint from the zeros of $g$ while maintaining uniform distributions on the choice of $g$ and $L$
