Couple of options:

1: They do. Once a decade or so, someone succeeds and promptly takes full advantage of the fact that nobody is going to connect the youth of 16 they now have the look of with the magus of 160 they were, assume a new identity and keep their gob shut. For maximal hilarity, this could explain the rumor about double witches - there is no such thing, but youthful witches and wizards with absurd powerlevels? Real, if rare.

2: The rite does not work well, or at all, for the old. Several options:

2a: The muggle mythos about the stone is not entirely off base. The creation of the stone requires a level or type of virtue exceedingly rare in people who have survived 150 +years in the wizarding culture.

2b: "Alchemy" is wizarding euphemism for "Tantric Magic", which is why all the books are restricted and while the spirit may be willing... >,) This also explains why Flamel only shares the stone with his wife - You can only help people you sleep with. This, of course, also rather nixes any of our heroes doing it anytime the next 5 years or so.

2c: For reasons similar to potions, the rite just does nothing for a caster over the age of 15.

3: Merlins interdict is screwing with the recipe- anyone wishing to make a stone has to do the research from scratch, and without an extant community of alchemical researchers, that is a project beyond the capability of any intellect ever born. - Flamels success happened in a context that no longer exists.