Analysis or flight test or both. This was the Aries-1 dilemma. Analysis, chiefly via FEA, showed longitudinal oscillations (in this case purely structural, since it was a solid-fueled vehicle). But the lack of FEA convergence couldn't be assertively attributed to actual mechanical properties or to the natural instability of FEA models. So the only option was a flight test, which was nominally successful.
As for problems cropping up in development, there's a maxim that says if you don't run into problems in development, you're not engineering hard enough. Which is a pithy way of saying that in order to make something of sufficient value to be notable or profitable, you have to bite off enough of a problem to make it unsure whether you're going to get there without difficulty. That was the spirit of Apollo. They were biting off huge problems, and the notion that they would get there without incident or serious difficulty is a gross misunderstanding of what engineering development is.