Clearly, due to the butterfly effect, storms will be highly influenced by quantum randomness. If we reset the world to 5 years ago and put every molecule on the same track, New Orleans would not have been destroyed in almost all cases.

No, no, no. The vast majority will have New Orleans destroyed, but in slightly different ways. Yes, weather is chaotic, but it evolves in fairly set ways. The original Lorenz attractor is chaotic, but it has a definite shape that recurs.