This in no way implies that everyone who teaches a student should be teaching those things.
Of course it doesn't. But if you make an assignment, you have a duty to grade it in a meaningful way.
(a) That's simply not true: students can learn a huge amount - both about how to wrote prose, and, often, about the subject they're writing about - from writing prose that isn't corrected by someone else. (b) One can make meaningful corrections to a student's work without correcting their writing, per se. One can correct the content, even!
I disagree with (a). Students can only learn so much about anything without having guidance from someone whose job is to teach them (this is the whole point of having teachers). Most people need help with writing; beginners, including most university students, need a lot of help. Even professional writers use writing groups to get advice and help from others.
(b) Just because it's important to correct content doesn't mean you can skip over significant problems related to structuring a paragraph or grammar. Outside of English class, suggestions don't have to be extensive, but they should be there. English teachers and professors should tear papers apart.
Do you criticise the English professor for not teaching arithmetic, too? Are that professor's students just supposed to figure it out?
No, I criticize a professor who makes an assignment and then doesn't correct it.
