As evidenced by this example: in such cases, the client is probably wrong. They'd be better off letting the user agent display the date picker that the user is used to.
Don't conflate the simple fact that there is an outright bug here with the presumption that the browser's date picker would be a better choice.
Even in the event that a browser ships a buggy date picker, at least it would be buggy on every website. The user then has a chance of understanding the bug and working around it to input the correct date.
OK, I'll use another tactic. Show me a single high-traffic travel website that uses the browser's default date picker. Travel websites are usually optimized out the wazoo because small increases in conversion rates make all the difference in this competitive business. Are you saying all those sites (the ones without bugs) are wrong?
The user is responsible for their browser. If they choose to use something that doesn't implement the spec and your site breaks, they sabotaged their own experience. That's very different to you sabotaging their experience.
This type of mindset is not conducive to creating a successful business. Or creating good websites in general.
Most users aren’t _choosing_ their browser, in fact they probably don’t even know what a “browser” is, relative to the “web” and the “internet”. If they’re using even a somewhat modern desktop/phone/tablet and things don’t Just Work it’s probably your fault, not theirs.