Preface: Obviously this is all nitpicking, for fun; you should use whichever system you like.
> Is it?
Yes, I'll stand by this one. Imagine it's January 1st, and I ask you to "pack for a camping trip some day in July this year" vs "…on the 10th of some month this year." While technically the latter narrows it to fewer days, the former is much more actionable because there's a lot less variability.
In my mind, it's similar to putting the main point up front when you're writing an email, and then clarifying, instead of giving all the context and working up to the request. The latter feels logical, like you're telling a story, but the former is so much easier to follow, because you don't have to re-interpret the context when you get to the punch line.
Here's a quick example: "I can't wait until you can come try it out. Such a smooth ride, tons of acceleration, and great handling. You're going to love riding my new tricycle." From a literary standpoint, it's a nice subversion of expectations, which makes it more likely to stick in your mind. But from a functional perspective — that is, communicate the ideas in an efficient, reliable, boring way — this would be much more effective, "I can't wait until you can come try out my new tricycle. Such a smooth ride, tons of acceleration, and great handling. You're going to love riding it!"
Back to dates. If you asked me, "Are you free on July 10th?" I would assume you mean the current year. When you ask, "Are you free on July 10th, 2025?" I do same thing, then hear the year and have to forget that and start over.
The real thing to optimize for is useful information up front. Can I start drawing conclusions as soon as you start giving information, or do I need to wait until the end? The programming equivalent is something like short-circuiting conditionals. I personally know I won't be able to take vacation in July this year, so I don't need to pay attention to the date once I've heard the month; but if the date came first, then I'd have to remember it in case the month turned out to be June, September, etc.
It's not a lot of overhead, and the date usually comes all at once in a row so it's not a very big difference either way, but I do think there's an objective difference.
> The real reason we find a system preferable is just familiarity
I grew up with mm/dd/yyyy. It's the most familiar/comfortable and, due to practice, the easiest for me to parse. I still think yyyy-mm-dd is better, for the same reasons as above (and because it's an international standard, and I like standards).