We present new advances towards achieving exponential quantum speedups for solving optimization problems by low-depth quantum algorithms. Specifically, we focus on families of combinatorial optimization problems that exhibit symmetry and contain planted solutions. We rigorously prove that the 1-step Quantum Approximate Optimization Algorithm (QAOA) can achieve a success probability of $Ω(1/\sqrt{n})$, and sometimes $Ω(1)$, for finding the exact solution in many cases. This allows us to prove a separation of $O(1)$ quantum queries and $Ω(n/\log n)$ classical queries required to find the planted solution in the latter setting. Furthermore, we construct near-symmetric optimization problems by randomly sampling the individual clauses of symmetric problems, and prove that the QAOA maintains a strong success probability in this setting even when the symmetry is broken. Finally, we construct various families of near-symmetric Max-SAT problems and benchmark state-of-the-art classical solvers, discovering instances where all known general-purpose classical algorithms require exponential time. Therefore, our results indicate that low-depth QAOA may achieve an exponential quantum speedup for optimization problems.