hedwards: Most of the time, the year isn't something we even bother to print. The year goes at the end because it's optional.
If it's not obvious, I wish we'd do it right and put the year at the beginning like the Chinese do. But, placing them from smallest to largest makes little sense other than to satisfy a compulsion. I genuinely can't think of any other reason for ordering them like that.
EDIT: Anyways, the reason why it's MM/DD/YY is because when we write the date out long hand, it's October 1st, 2015 and such. Month, day, year.
It`s the same reason you mentioned, because, like year, sometimes month is not worth mentioning.
"- when have you sent the report?
- on the 5th. "
It's clear that the report have been sent on 5 October this year. And if it was said 28, then it should be clear that it was 28 September this year, because otherwise the month would have been mentioned.
So, we sometimes use DD/MM or DD/MM/YYYY because, as you said, we write the date out long hand, but after seeing the first numbers (the day) I already know what I`m looking for.
The most annoying thing is when I see something like 05/06/2015, I don't know if it`s 05th of June (how it should be) or the one who wrote it is frigging american, and it`s 06th of May (no offense to yankees :P).