I dimly remember trying this five years ago and it being a bit of a mess. IIRC it was that date picker widgets in browsers presented in a different format to what the Accept-Language header suggests (a killer when you're trying to render static and editable dates in the same table), or users were confused seeing dates in their preferred format after being linked from a site that disregards them, etc. Perhaps the situation has improved by now.
Seems like requiring the date author to think about widgets is too much of burden. Better to just leave it as text and have an annotation layer that optionally says "this substring is a date in format X with timezone Y" which viewers can optionally re-render according to their preferences.
That way if neither option is exercised, it's still just text.
Don't blame the other culture, blame your inept communication medium.