Luckily, Intl.DateTimeFormat [0] seems to finally allow a reasonable degree of i18n in the browser. If you can live with the limited Safari availability, of course.
[0] https://developer.mozilla.org/en-US/docs/Web/JavaScript/Refe...
That'd quickly get awkward. I live in Norway where the default format is DD/MM as opposed to MM/DD used in the US. This works fine for us in general, we know to expect the first one when we read something in Norwegian and the second one when we read something in English. If a computer formatted a date based on a default preference and considering that most websites probably don't specify the lang attribute, I think it'd get messy quickly. Also, different date formats will lead to different sentence structures making sense. For example you might say "ever since 9/11 something something" but you wouldn't naturally write "ever since 11. September 2001 something something" (you'd write "ever since the 11th of September 2001 something something") and likewise you also wouldn't naturally write "ever since 2001-09-11T08:46:40-0400 something something".
On more serious note, such functionality is meant for formatting freestanding dates/times (eg. contents of table cell labeled "last modified") not dates in flowing text.
What you suggest would make that difficult if not impossible. Already it is unnecessarily hard, as JS Date object automatically uses local tz and cannot be coerced otherwise.