When you ask someone “what day is it today?”, they usually have a handle on what month it is and just need the day.

You’re still allowed to exclude implied information, no matter which method of dating you want to go with. You can just say “the 15th”.

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.

I can’t speak for you, but for me I am making plans, sorting, and scheduling every single day.