Earlier quoted context omitted.
With languages that are not written phonetically with wide regional variations in pronunciation (English), I think it makes sense to not care as long as you're understood. Might as well demand that Japan be pronounced Neep-on or Cuba Cooba and not Cyooba.
I always liked the BBC's approach -- English words and well settled loan words are allowed to have regional pronunciation, foreign words go via the pronunciation unit. So news readers pronounce (or try in the case of that Icelandic volcano) locations, names etc as they are intended to be. Names are personal, it's offensive to mispronounce them. Make the effort, even if it involves struggling a little with sounds that…
Take Iran for example. It's rendered often in English as [ɑɪ.ɹæn] (eye-ran). What it "should" be is [i.ɾɑn] (ee-rahn). Fortunately, all the sounds in [i.ɾɑn] are ones that occur in most varieties of English (with the exception of the [ɾ], but that's minor), so most speakers are able to accommodate without issue.
Now compare that to Shanghai: its English rendition is [ʃeɪŋ.hɑɪ], and its Mandarin pronunciation is [ʂâŋ.xài]. Most Englishes lack the voiceless retroflex fricative [ʂ], and the same for the voiceless velar fricative [x], and to make matters worse, Mandarin is tonal, which puts the correct pronunciation of this word squarely out of reach for almost all English speakers.
It still makes sense to insist that people try as hard as they reasonably can. Perhaps to say that speakers should get as close as possible—in the case of Shanghai, for example, maybe this would mean [ʃɑŋ.hɑɪ] instead of [ʃeɪŋ.hɑɪ].