Most references seem to state that Nash showed every symmetric game has a symmetric equilibrium point, but as far as I can tell from Nash's paper, he actually showed the much more general statement that every finite game has a symmetric equilibrium point.

I realise the references wouldn't be wrong, but why don't they state the more general result?

Equilibrium Points in n-Person Gamesfrom 1950 only contained a proof that an equilibrium exists.Non-Cooperative Gamesfrom 1951 (essentially the published thesis) contains a proof that every game has a symmetric equilibrium. – Michael Greinecker Apr 14 at 17:01