Indeed, one of the stories at qntm features discovering something like the Fountain of Youth, and describes the organizational nightmare of letting a large enough proportion of mankind use it.

Harry Potter, however, if he makes himself immortal, will have a lot of time. Even if a lot of people die of old age or illness before he can reach them, he can still reach a lot of people given enough time. He would also need to figure out some efficient way of space travel and terraforming, otherwise he could cause overpopulation.