Not one language has speakers who will naturally pronounce "Shinjuku" exactly correctly either - no two languages are alike, and it's normal and healthy to have pronunciation differences between languages natively written with the same alphabet (e.g. English speakers will naturally mispronounce many Spanish words - no-one would suggest this means Spanish spelling should change). Plenty of languages (e.g. Irish or Polish) naturally read "Sinzyuku" as something closer to the correct pronunciation than "Shinjuku", which is the only reasonable way to compare.