This "verbal, intuitive character" is its biggest flaw: it makes grammar look like it is logic, while it is not. That's how you get "cogito ergo sum" instead of "cogitationes sunt", or the ontological argument.
Let me clarify the area where I believe it is useful: the study of term logic helps prepare the minds of non-philosophers to reason, speak, and write clearly about concrete but often ill-defined matters of everyday life. While it lacks rigor, it forces them to consider the implications of their words and those of others, and how those words operate to express proof and disproof. The study of symbolic and other more formal systems of logic, while essential for more rigorous reasoning, requires too much time and application to achieve similar results, because their mode of expression is too removed from the experience of non-philosophers and especially non-mathematicians.
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).