if I ask if you’re free on “the 15th of March” vs “March 15”, the first example is slightly jarring for your brain to interpret

Sounds like you’re just used to it being said the opposite (read: wrong) way. If you told someone in my country March 15th, it would be just as jarring to the listener.

at first it hears “15th” and starts processing all the 15ths it’s aware of, then “March” to finally clarify which month the 15th is referencing.

not in daily use. When you ask someone “what day is it today?”, they usually have a handle on what month it is and just need the day. For making plans, it’s only if you make them way in advance that you need the month first, which would be sorting and scheduling, not daily use.