That there are people who don’t think nihilism is a problem, and that a time may very well come when no one feels nihilism is a problem, is precisely why nihilism was a problem for Nietzsche.
Nietzsche prized human greatness above all else. To achieve human greatness, he thought, there must be problems and there must be people who care about them. That’s because greatness results from overcoming problems.
What all problems have in common is suffering. Suffering, therefore, is good, not in itself, but because it’s a necessary condition for greatness. But the modern world, Nietzsche thought, is in the process of eliminating suffering by creating a world of abundance, security, and comfort.
For reasons I won’t go into, Nietzsche believed that eliminating suffering requires the elimination of meaning as well. However, eliminating both meaning and suffering requires human beings for whom the absence of meaning isn’t a problem – since otherwise they’d suffer from it, and then suffering wouldn’t have been eliminated.
Under those conditions, there would be no more human greatness, because no one would suffer and there would be nothing to overcome. Nietzsche hoped that a certain type of person will continue to suffer from the absence of meaning, approach it as a problem, and achieve greatness by overcoming it.
In short, nihilism is a problem for Nietzsche because if we become used to it, there will be no more human greatness. The question then becomes: Why is the absence of human greatness a problem?
That, I think, is something Nietzsche believes you just either get or don’t get, depending on your personality. To someone with a robust love of life at its most intense, the value of human greatness is self-evident. To someone who prefers security to intensity, the absolute value of human greatness is somewhat less than self-evident.
From Quora: Why was nihilism a problem for Nietzsche, when there are other people who don’t think it’s a problem at all? This is the answer from a Professor of Philosophy at Berkley.