Most normal people decide at some point that math is not for them in order not to embarrass themselves by claiming competence in a domain in which they are not confident at all. And hence, compared to nerds or seriously talented individuals who can confidently claim this status, they never spend much time on math and eventually seriously lack skills even in simple logical reasoning. This is basically math anxiety. Nerds, on the other hand, are incentivized to gain social status via math skills as a potential escape of their low status. I think this mainly explains the author's observations that these two skills are not correlated very much, basically an introversion vs extroversion polarization based on social expectations and incentives. Math is also intimidating, so I'd imagine someone with some experience in it also develops overall higher inhibition and hence is a worse verbalizer.
There is a huge literature on the relation between logical reasoning and verbalizing, which the author sadly ignores.