“A man's grammar, like Caesar's wife, should not only be pure, but above suspicion of
impurity.” said Poe.

If Poe actually 'spoke' those words, I contend that he wrote and edited them prior to speaking them. Grammar's performance in any spoken language is seldom pure and never above suspicion of impurity.

I believe I understand your contention that a native speaker is able to 'clean up' his idiosyncratic spoken language when teaching, whereas a non-native speaker is unable to improve his English speaking ability in the classroom. Perhaps, on the margin, that is correct. But I know some non-native teachers whose spoken English is superior to that of many NES FTs in China. Granted, this is not the norm. I'm thinking of one Dean at a Chinese University who was born in China, graduate college in China, but then lived in Europe for 15 years and graduated Oxford with a PhD. Her spoken English is grammatically superior to my own.

Pronunciation is a separate issue. As a U.S. citizen, I am more tolerant of 'sloppy' pronunciation. I have no idea if a well-educated resident of Liverpool can 'normalize' his pronunciation for the classroom.

Personally, I never attempt to shed my New York vowel extensions or add final stops. Forgetaboudit.