Not always. Some problems are non-deterministic (their solution). Apart from that, some problems are, as you say, so sensitive to changes in initial conditions, that most solutions are too localized.
But there are cases where quantum computers can provide insightful results, that might shed light on different approaches to solutions.
Another point to consider is the use of Numerical methods in chaotic systems. Some methods are more optimal than others, at the cost of accuracy. With quantum computers, computation time decreases by a lot (acc. to theory), which may allow more accurate calculations, leading to a better understanding of the more difficult chaotic systems.
To clarify: Quantum computers might not be able to give an analytical solution (even to problems that might have such solutions), but a more accurate approximation can often lead to a new understanding of the problem, which is a way to handle problems.