To reformulate the question bit in more familiar terms:
"Which type of argument is more robust to error: deductive or inductive?"
(I'm not overly familiar with Nietzsche's work so I'm going to cheat and assume that since he stands against holding ideas a priori, he tends to reason from an inductive stance. I assume that's what you mean by "social analysis". Plato's Republic also does what I'd call "social analysis", but as you imply he tends to rely on first principles.)
To answer: it depends. I disagree, for instance that the inductionist's "entire analysis doesn't collapse if he's wrong". Afterall, empirical evidence suggested that atoms were the smallest possible unit of matter (hence the name) until further evidence showed that smaller entities existed. As time goes on, the Atomist Theory becomes increasingly incorrect as more data is gathered.
Meanwhile, many of the practical conclusions of Plato have been shown to be wrong in the sense of accuracy, but his works are still very relevant today despite being built on assumptions that are no longer commonly held. Many of his ideas can be recast into contemporary terms without disrupting his reasoning process drastically. His theories of social stratification don't have much traction today, but his idea that society ought to be governed by rational principles holds up very well.
Perhaps a better criteria for robustness is that general claims tend to be more robust in the face of error than specific claims, but specific claims tend to be more useful when correct. "Diamonds are hard" is a general claim likely to survive any new information, but "diamonds are the hardest material" is a more specific (and useful) claim that may not survive new information. Note that both these claims are inductive based on observation and not a priori assumptions.
Not at all an answer to your question, but a personal opinion: Nietzsche (like Tolstoy and Pascal) seems far more a product of his time than Plato. He seems like a niche philosopher who appeals to specific personalities and cultures more than to others. His way of looking at the world doesn't appeal to me and therefore I have a difficult time following his lines of reasoning. Pascal is the opposite for me: I get where he's coming from although I know many other people don't. Meanwhile, both Plato and Aristotle, who come from opposite sides of the deductive/inductive spectrum, seem to hold up well from many people. I have no direct evidence for this assertion, however.