That isn't at all what the central limit theorem says. The whole point is it holds independent of the actual shape of distribution of the population. You could use the same argument to say social security numbers are normally distributed.
One way to explain things like height being normally distributed is that there are a bunch of independent factors which contribute, and the central limit theorem applied to those factors would then suggest the observed variable looking normal-ish.
One way to explain things like height being normally distributed is that there are a bunch of independent factors which contribute, and the central limit theorem applied to those factors would then suggest the observed variable looking normal-ish.