Is key size the only barrier to the adoption of the McEliece cryptosystem, or is it considered broken/potentially vulnerable?
The huge key is definitely an issue. Another is the lack of standardization or recommendations. Should you use OAEP with McEliece, or some other padding? What parameters are actually secure? And so on.

Part of the problem is that, while it has been around since the 70s, it was not considered particularly interesting until quite recently—so it probably hasn't had that many research years spent on its security properties. And so far as I know, it's never been deployed in the real world.

Other potential candidates for post-quantum crypto include NTRU and HFEv-, which are fast and have small keys. Unfortunately, both are patented. If you're interested in post-quantum cryptography, I would highly recommend picking up Post-Quantum Cryptography by Bernstein, et al, which is a good survey text.