Ok, let me rephrase that: See how 4th of July feels natural for you to say or hear?

That’s because you’re used to it.

That’s how it is for the rest of us using DD-MM-YYYY, but for all dates.

Because that’s what we are used to. And yes we use it in everyday life and in every conversation.

That is all there is to it.

Say when is that independence day of yours?

There is nothing ‘natural’ about mm/dd/yy, it is just what you are used to - end of story.