Metrizability dual of countable discrete group

Question

Let $G$ be a discrete group. It is well-known that its dual $\hat{G}$ is compact. I have read in a couple of places that $\hat{G}$ is metrizable if and only if $G$ is countable. I know how to easily prove this using the Birkhoff-Kakutani theorem (i.e. a group is metrizable iff it is first countable). However, I am wondering if this theorem is really needed for the proof or if there is a different and easier proof which avoids the use of a theorem as strong as Birkhoff-Kakutani?