Why do Americans want to know the month first and the day second?
I'm sorry but it doesn't make sense TO ME. Based on what I was taught, regardless of the month, I think what matters first is to know what day of the month you are in, if at the beginning, in the middle or at the end of said month. After you know that, you can find out the month to know where you are in the year.
What is the benefit of doing it the other way around?
I'm not Mexican, but this reminds me of a Mexican ranchera that says "No cabe duda que es verdad que la costumbre es más fuerte que el amor" (There is no doubt that it is true that habit is stronger than love).