I don't think that the N-body problem exactly qualifies as a black-box type problem, but I think I understand your argument. For some problems, particularly certain types of complex adaptive systems, I can't imagine a better approach than simulation - that is, I can't imagine an analytic solution to the problem.
Computer time is now cheap and so it's easier and vastly less expensive to solve certain problems numerically than to attempt analytic solution. My perception is that in the 60s and 70s the Russians were doing much better at developing math than the US mainly because they didn't have computing resources and they had no choice other than to get smarter.
I think it's a situation with perceived cost-benefit. People think, "Why spend all these resources when we can just do it numerically?" OTOH, it would be good for N-body even if we had a proof, say, that there is no analytic solution. That way we'd know up front if it were a waste of time.
Somehow I think we're going to be waiting for the next Newton, Gauss, or Erdos for an analytic solution to N-body. In the mean time there's all these underutilized computing resources lying about.