Attention is significantly focused on parts of speech, proper punctuation, grammatical rules, basically the foundation of a style guide presented as a bit more than just a style guide.

I expect this varies significantly depending on the setting in which English is taught, but this was far from my experience in high school English. In my experience, grammar is hardly taught in English classes at all. Instead, the whole class is basically focused on what you describe as the 'other side': literature.

The problem I found, though, and that I hear again and again from pretty much everyone else, is that English teachers have a tendency to over-analyze and even to impose their own preconceived notions onto it, to the detriment of those who are forced to sit through it.

I think you and I may just have to agree to disagree here, but again, this is far from my experience as both a student and an educator. I think most English teachers try to teach students how to close read texts, and, as such, they offer examples of close reading. However, they are usually very happy to get students' readings of texts (even if students are sometimes very hesitant to give such readings).

I like your suggestions of Wu Tang Clan and Eminem in literature classes. I've also found that fan fiction can work in interesting ways for this. Introducing texts like these into the classroom can be tricky, but I think it can work really well for the students if it's done carefully.