I am struggling to convey this, so I'll have to think about it more.

For now, though: I do think that differences in the initial conditions would be propagated by adaptive individuals and institutions (rather than smoothed away). That should lead to bifurcations and path dependencies that would generate drastically different outcomes. Enough that averaging them would be meaningless.

Why do you think repeating it many times would converge? Are the statistical limit theorem conditions really met? I don't think so..

None of this really explicitly says that you wouldn't be able to at least figure out the sign of the change. It might be computationally intractable but qualitatively determinable in special cases.