There is very little relationship between the structure and grammar of languages and how they are written. Bear in mind that most languages that have ever existed were unwritten, that some languages have been written in more than one writing system, and that native speakers of a language almost invariably learn to speak before they learn to read and write.
Whatever might be their view for or against Chomsky's Universal Grammar, virtually all linguists agree that language "happened naturally" in some sense, whereas writing was an invented technology, that came along only in the last few thousand years. It is conceivable that the writing system of a language could have been devised with reference to the grammatical structure of a language, but I'm not aware of any examples. (Sanskrit is the only example we have, as far as I am aware, of a pre-literate tradition of grammatical analysis; and while the Devanagari script is ordered on phonetic principles - unlike the arbitrary arrangement of European alphabets - I don't believe there is anything particular to Sanskrit grammar in its construction and use).