When we're saying a date out loud, we Americans tend to say, "September 21st, 2023," whereas I've noticed many of the Brits I've worked with have tended to say, "21st September, 2023," sometimes with an "of" after the day. Those spoken word orders were carried over into the written form, is my suspicion.
If the the month was unclear you might reply with "21st of what?" but you'd never expect an answer of just "September" to the question "What's the date?".
My assumption was it's due to an attempt to make the written date format closer to the HH:MM, and they just forgot about the years, with the spoken "Month, DayOfMonth" coming later.
Do people in the US generally say "the 4th of July" or "July, 4th" more? That might hint at an origin.
As for why the month comes first, I get the feeling we Americans care about precision up to a month and less so about the numerical day such as "When does school start?" with an answer of "School starts in September". Obviously people need the precise date if you actually have a student going, but in a lot of conversation, people are just looking for the month. I can imagine that being even more true in earlier times.
We also peg things to months such as our elections being the second Tuesday of November or Labor Day being the first Monday of September. Note that in those phrases we are putting the day kind of first, but not in a particular numerical way. The month is what sticks out as a rough guesstimate of the time period.