The Goldreich-Goldwasser-Micali Construction with bad PRGS
I know I'm answering my own question, but there's actually quite a nice worked example which completely refutes my question. I thought I'd share.

Suppose I use the GGM construction with the underlying PRG being RC4. 

RC4 has a lot of problems, one of which is the fact that the second byte is basically twice as likely to be zero than it should be: $2\over 256$.

For longer key-streams, the biases settle down and the cipher can be used relatively safely. Google, for example, uses RC4 for in SSL. 

However, the GGM really punishes RC4. Every stream that RC4 generates is short. An attacker can use this to distinguish the resulting PRF from random very easily. Here's how.

The attacker requests that you compute your PRF on values of his choice. He can craft those values so that the tree that is navigated in GGM always returns the left hand side of the expanded bit stream. These are the first bytes out of RC4.

He can then simply look at the second byte of the result of PRF(x) and count the number of times its zero. After a pretty trivial number of queries, he's clearly able to distinguish the PRF from a random function. 

We're now in a position where the GGM has actually has poorer security than the underlying PRG! This nicely refutes my question.

The message is that a bias in the PRG should be assumed to render GGM insecure.