I'm a big Chomsky nerd, Chomsky fan, and card-carrying ex Chomskyan linguist. I hate to break it to you, but not even Chomsky thought that the Chomsky hierarchy had any very interesting application to natural languages. Amongst linguists who (unlike Chomsky) are still interested in formal language classes, the general consensus these days is that the relevant class is one of the so-called 'mildly context sensitive' ones (see e.g. https://www.kornai.com/MatLing/mcsfin.pdf for an overview).

(I suppose I have to state for the record that Chomsky's ties to Epstein are indefensible and that I'm not a fan of his on a personal level.)