Is is the device display language, the keyboard input language, my geo location, my browser language, my legal location, my browser-preferred website language, the language I set last time, the language of the domain (looking at amazon.co.uk), the language that was auto-selected last time for me on mobile or... something else entirely?
And for the rest of the users who have no idea about locales, using whatever locale they have on their computer might be technically incorrect for some of them, but at least they're somewhat used to that incorrectness already, as it's likely been their locale for a while and will remain so.
This is the equivalent of requiring all your text to be in Esperanto because dealing with separate languages is a pain.
"Normal" people never use YYYY-MM-DD format. The real world has actual complexity, tough, and the reason you see so many bugs and problems around localization is not that there aren't good APIs to deal with it, it's that it's often an after thought, doesn't always provide economic payoff, and any individual developer is usually focused on making sure it "looks good" I'm whatever locale they're familiar with.
- Use localization context to show the right order for the user
- Display context to the user that makes obvious what the order is
- Show the month name during/immediately after input so the user can verify
(And yes, always accept YYYY-MM-DD format, please.)