> This "verbal, intuitive character" is its biggest flaw: it makes grammar look like it is logic, while it is not.

I don't know if I would say it makes grammar look like logic. Rather, it holds that grammar has a logical structure. You would still need to regiment your arguments in many cases if you wanted to explicitly employ the formalism in question, but this is viewed as faithfully bringing out the essential logical character of a proposition latent in the grammar, not as a replacement of the original proposition with something else. Recall that FOL is completely indifferent to grammar (though apparently this hasn't stopped linguists from using FOL to analyze natural language, something Sommers sees as resulting in a distorted view of natural language).