I don't doubt that the people you mentioning are the age they say they are, but even when there are multiple documents verifying somebody's age, they all take get data from a single original document. Apart from this I see things the same way as you've detailed.
The thing is - what I'm talking about is _not_ coming from a single original document. Yes, the birth certificate is a single document, but there's other documentation where children are mentioned. I looked up everything I could find about the 99 year old gentleman I mentioned, and because of certain "interesting" things about two of his grandparents the family had been traced by multiple sources before and through his birth. Newspapers were around, and photos too. The situation is different now than when people claimed to be extremely old back when the rage about super old people in certain east-European / west Asian regions began. Now we're talking about people born way into the 20th century, which means that for many regions in the world there aren't many options to fake it.