It's funny how the Germans and Dutch (rightfully) ridicule Americans for writing dates in middle-endian order like 9/11/2001, yet they say numbers with the decade flip "two and forty". That's just as ridiculous.
MM/DD/YYYY is simply a direct transliteration of spoken English, which makes it easy to read and write dates. In other languages, the spoken version is little endian or big endian, and the written version aligns accordingly. (At least for the languages I know.)
We would normally refer to that as September 11 because it's much more talked about in the US, and that's the phrase used there.
Any other dates will likely be in the same order as written. For instance, the rhyme for bonfire night is 'remember remember, the fifth of November'. I believe that many in the US also talk about the fourth of July, rather than July fourth, so it's not like English has the hard-and-fast rule you were proposing.
Interesting. Would you choose "three July twenty fifteen", "July third twenty fifteen", or "the third of July twenty fifteen" (substitute "two thousand" for "twenty" if you like)? Assume someone has asked you the date and you're responding out loud.
It took me a while to figure this out because actually it's quite rare to speak a date including a year without reading it - most spontaneously-spoken dates are this year (so the year is implied) and for a read dates, I'd probably say whatever was written.
The clincher was how I'd say my birth date, which would be of the form above.
I'm not claiming to be the definitive British English speaker, though! ;)
...and as another poster commented, it might depend on context - for example, "September 11" is often used in British English because it refers to an American event.
Americans read and write dates in middle endian. Germans and Dutch only read numbers with decade flip, they are written as usual. Furthermore with "two and forty" there can be no confusion since it's not "two and four", so it's clear that "forty" refers to the tens position and "two" to the units position. It is of course not ideal, but not nearly as much cause of confusion as the middle endian dates, because there's simply no way to know what 9/11/2001 means.
At least they (we) stay consistent between 13 and 99, while a certain other language elects to switch the flip at 20.
I think we should all take a moment to admire the francophone Swiss for boldly dropping much of the madness that is french counting. (Yes, I am looking at you, quatre-vingt-dix-neuf!)