When thinking about crossover its important to think about the fitness landscape.
Consider a hypothetical scenario where we are applying a genetic algorithm to find a solution that performs well at 2 tasks. This could be from Franck's example (moving and shooting) for an AI, or perhaps it could be predicted 2 outputs in a genetic machine learning scenario, but really most scenarios where GAs are applied are synonymous (even at solving a single task, there may be different aspects of the task to be addressed).
Suppose we had an individual, 1, that was performing reasonably well at both tasks, and we found a series of mutations which produced 2 new individuals, 2 and 3, which performed better than Individual 1 at tasks 1 and 2 respectively. Now while both of these are improvements, ideally we want to find a generally good solution, so we want to combine the features that we have been found to be beneficial.
This is where crossover comes in; by combining the genomes of Individuals 2 and 3, we may find some new individual which produces a mixture of their performances. While it is possible that such an individual could be produced by a series of mutations applied to Individual 2 or Individual 3, the landscape may simply not suit this (there may be no favorable mutations in that direction, for example).
You are partially right therefore; it may sometimes be the case that the benefits of crossover could be replicated with a series of mutations. Sometimes this may not be the case and crossover may smooth the fitness landscape of your GA, speeding up optimization and helping your GA escape local optima.