In Chinese, its also Month, then day.

Its always "五月 三十一号" (May 31st), never "三十一号 五月" (31st May)

When saying a day in history, the full format is used: 一九六九年 七月二十日 (1969 Year 7th Month 20th Day), but when you use it during every day life to refer to a day in the same year, the year is omitted. If you are referring to the same month, the month also get ommitted.

TLDR: Not unique to "Americans"

For no other reason than to be different and contrary. Metric system anyone?

We don't want or care about it. Its just the way its always been over here. Same with fahrenheit and the imperial system. I know this is a little exaggerated but imagine asking a Japanese person why they dont just speak english because its the worlds most spoken language. Its one of those things other countries like to pick on us for that I think is a little stupid. Instead you should be dunking on us for putting an orange party clown in the whitehouse. Again. But then again there are some exploitative processes that helped make that happen like gerrymandering. My point is that our shit healthcare and the majority of the things that we are laughed at for is out of our hands. Even the fat jokes can be somewhat blamed on the fact that the availabilty of cheap unhealthy food with a large portion size is greater than pricier organic options and lots of the poor tend to take that route. I'm not saying our majority is completely blameless but these are factors to consider.

Edit: following your Edit: thank you for the clarification

Linguistics

In UK English, it's considered proper to write "the 6th of March" as "6 March" and sometimes read as "6th March" which can be jarring to Americans as their shorthand is "March 6th" and when "6(th) March" is encountered in written form, it's expanded to the full "6th of March" when spoken

That doesn't mean this won't be yet another feature American English absorbs from UK English but right now flipping them in speech requires a few extra syllables and people are lazy

I don't think there's any real logic behind it other than the founders of our nation post the war of independence did not want to have any real similarities between America and Britain.

All display of time should follow this format:

Chronon.PlanckTime.Yottayear.Zettayear.Exayear.Petayear.Terayear.Gigayear.Megayear.Kiloyear.CosmicAge.GalacticYear.Epoch.Eon.YourMom.Era.Aeon.Megaannum.Millennium.Century.Decade.Year.Month.Day.Hour.Minute.Second

The month first is best because consider what happens if a message gets cut off. You might get: "You'll be flying to New York on the first of ..." or "You'll be flying to New York on June..."

The first message doesn't tell you anything useful. Do you need to buy shorts or a parka? Do you have months to prepare or are you leaving in a few hours? Could this be an april fools joke? It's a 1/12 chance. Totally useless.

Second message, sure the details are unclear but at least you know what to pack and that you need to hurry about getting the rest of the message.

Context clues.

I say June 2nd of 2025

I type 2025-06-02

Handwritten it's 2-June-2025

I'm from before 2000 and the turn to years being so small broke me, it used to be so clear which number was the year with just 2 digits, and day, month, year is sorting from smallest unit to biggest, it has logic. But then for awhile you could have a 04, an 05, and an 06 and I was working with other countries, it wasn't at all clear which was year month or day, so I started sandwiching the month in the middle as a word when handwriting dates and using 4 digit year, and year month day sorts like a dream for filenames.