English pronunciation belongs to the study of English along with phonetics, syntax, semantics, etc.and true, those things are inseparable. I know of some people whose mere interest is to write and read in English - nothing more, nothing less. That goes for some of my realtives still living in Europe. One of them is a Physics professor whose writing skills in English are spectacular but if he were to say something in English, you would really end up scratching your head....but all he wants is to read and write in English.
Pronunciation has never been a big thing, never really mattered one way or another.
Just a little digression...
Teaching students is different. Getting a head start on proper pronunciation is important but putting kids in booths with head phones on to practise minimal pairs (or having you do this with them) seems a bit much. I also don't think it's a good idea. This comes with time and getting a handle on the very basics in grammar and pronunciation should be a priority. Without that, hell with phonetics and whatever else. If the student doesn't know how to express himself by using horrible grammar, then what's the point in polishing his sounds or pairs?
When I used to teach, pronunciation wasn't so much of an issue but certain parents used to put a lot of pressure on me for all kinds of things, including not giving his/her kids enough chance to participate in class. Turns out the kid was way in the back row somewhere working on his math homework.