I understand the point entirely. I said I agree with it.
But at the same time, "Fourth of July" should not be used to infer anything about how Americans say dates. It doesn't help prove that both orders are used for dates.
In the real world, some people say day-month and some say month-day, and the holiday is called "Fourth of July". In a counterfactual world where everyone switched to month-day, the holiday would still be called "Fourth of July". So you can't use "Fourth of July" to demonstrate what people actually use for dates.
I'm not saying Americans say dates like that, I'm saying it's a reason they might consider that other people, or Americans once-upon-a-time, say dates like that.
But at the same time, "Fourth of July" should not be used to infer anything about how Americans say dates. It doesn't help prove that both orders are used for dates.
In the real world, some people say day-month and some say month-day, and the holiday is called "Fourth of July". In a counterfactual world where everyone switched to month-day, the holiday would still be called "Fourth of July". So you can't use "Fourth of July" to demonstrate what people actually use for dates.