False. There is no theoretical or comprehensive empirical study I've ever seen to suggest this is the case. At best, in some scenarios (e.g., multiplex, ordered subset selection) crossover will help speed up the search because it is effectively assuming it's separable, meaning different parts of the problem can be optimized independently. So if you're trying to get all 1s in a bit string, you could imagine separating that into two parts: maximizing the first and second halves of the bit string, then combining the best individuals you would get an optimal answer to the original problem. This is why, for instance, that crossover rarely helps in evolving neural networks; their weights' non-separable nature makes it difficult to select a meaningful subset of the network.

> Crossover is somewhat helpful against local optima because it allows the algorithm to try combinations of innovations from different solutions. Depending on the fitness landscape, this can be either helpful or useless. Regardless, a GA will, on any non-trivial problem, almost certainly get stuck on a local optimum, which is a very important point for users of GAs to realize.

Also false. I'll especially take note that this is not the case if you're using a less-than-naive EA. One great example is CMA-ES, which is particularly well suited for handling non-convex, deceptive optimization problems (and uses no crossover).

> But what is not the case is that mutation prevents crossover from hitting a local optimum. Crossover by itself basically just won't optimize at all

Again, not true. Consider multiplex. You start with a bunch of random bit strings. IF you have 1s in all indexes somewhere in the population, it's possible (though unlikely) that the algorithm will find the optimal answer. The issue that arises is when you select against the last 1 bit in a certain index in the population; it's now lost forever and thus the GA will at best reach a local optima (optima in the sense of it's the best it can now do with future populations).