Is key size the only barrier to the adoption of the McEliece cryptosystem, or is it considered broken/potentially vulnerable? A recent paper showed that the McEliece cryptosystem is not, unlike RSA and other cryptosystems, weakened as drastically by quantum computing because strong Fourier sampling cannot solve the hidden subgroup problem.

However, in 2008, new parameters were proposed after the old ones were found to be vulnerable.

So my question – as someone who hasn't much knowledge on this particular cryptosystem – is: Why isn't it in greater usage?

It would seem that if the promise of quantum computing is fulfilled, we may soon have need for cryptosystems not vulnerable to it. So, is the requirement for a large matrix (and therefore a large key) the only barrier, or are there other concerns/potential weaknesses? Or is there a better candidate for a post-quantum public key system?
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.