Someone who craves fame and wants to be remembered after death doesn't think this through. The people who remember him will soon die too. Then the people who come after them will also die. Eventually all memory of him will disappear completely. It has always worked this way — people admire someone, then die, then get replaced by others who also die.
But let's say the people who remember you could live forever, and your memory could last forever too. What good does that do you? I won't even ask what good it does after you're dead. What good does praise do you while you're still alive? The only use is for practical reasons — what we might call strategy or management.
He who is greedy of credit and reputation after his death, doth not consider, that they themselves by whom he is remembered, shall soon after every one of them be dead; and they likewise that succeed those; until at last all memory, which hitherto by the succession of men admiring and soon after dying hath had its course, be quite extinct. But suppose that both they that shall remember thee, and thy memory with them should be immortal, what is that to thee? I will not say to thee after thou art dead; but even to thee living, what is thy praise? But only for a secret and politic consideration, which we call οἰκονομίαν, or dispensation.