I am a fairly militant, pronouns-in-my-bio (he/him, cishet) post-colonial feminazi and I think so-called politically correct language is generally a good thing. But I also have a degree in linguistics, and I have to say I don't think gendered grammatical categories (that is distinctions in pronouns, noun classes, verbal conjugations, etc. that align with cultural concepts of binary gender) have any effect on gender relations in a culture at all.
The (Classical) Arabic language is fairly unique in the Muslim world in expressing gender grammatically in several ways. For instance, in addition to a he/she distinction, there are separate second-person pronouns ("you") for use when addressing men and women, as well as matching, gendered forms of the verb. That is, you can't speak to someone without encoding their gender grammatically, sometimes even several times in the sentence.
Compare this to Turkish and Persian, where gender is not expressed at all except for some words. There are separate words for man and woman, and male and female family relations but that's about it. There's one third person pronoun, so you can write love poems that are completely gender-ambiguous (indeed, much classic Persian poetry can be read as being either about love of another person or about devotion to God). As far as I understand, Turkish even has many names that are gender-neutral. Bahasa Indonesian and its related languages are also free from grammaticalized gender.
And while there are many differences between all aspects of society in the Islamic world, including gender relations, if gendered language was an important factor, I would expect them to be much more aligned with linguistic boundaries, and I have never seen even the tiniest shred of evidence for that.
Bahasa Indonesian is quite interesting, by the way, in that it and languages of that family (that Austronesian languages) have interesting ways of encoding social information grammatically. You might have more than one first person pronoun ("I") that are appropriate in different contexts, and the relative social status between the speaker, the addressee, and the subject are encoded in various ways. I can only assume that Indonesian letters to the editor about the lack of respect that young people have for their elders are full of just-so linguistic argumentation, and that the inevitable clashes between generations have elements of linguistic reform. But at least when trans people gain recognition there, they won't have to argue about pronouns.