Why does anyone need a site for this? I can pick my own random dates just fine. And ain't nobody got time to write down which random date they used on which site on the off chance they need it.
The website actually recommends against using the same date for everything. I think that was the point of the website rather than giving a random date. Namely, promoting not using personally identifiable information such as date of birth. I suggest thinking of the random date display as a sort of practical joke.
Also you're right that no one writes down their fake date of birth -- that would be weird. As an advice that wouldn't make sense, but what if we consider the case where people "don't care" and always chose "default" date of birth like "Jan 1, 1999"? That would be the same on all websites.
