Have you tried automatic programming? For one heuristic optimization project, I automatically derived arithmetic expressions to match an unknown function, using a wide variety of methods including hill-climbing. Many methods started out more efficient than genetic algorithms - at one hundred thousand evaluations or so, simulated annealing was doing best. But by one million, all other contenders had levelled out, while the genetic algorithm was still doggedly finding better answers. The fact that it maintains a population and exchanges working code between its members turns out to be extremely effective for hard problems.
Yes, I have tried automatic programming in the past, but my results were the opposite. Hill climbing universally does better. Of course you need to be reasonable and restart the search once it gets stuck in a local maximum. If you don't do that then GAs obviously do better since they start off with a whole population rather than one candidate. So the reason GAs do better has nothing whatsoever to do with its essential difference to hill climbing (the crossover). In fact if you just take an ordinary GA and disable crossover, and you let each individual in the population have exactly 1 offspring, it will usually work better! This is because crossover & the breeding rules of GA tend to remove diversity in the population.
By the way, in my automatic programming programming experiments hill climbing hardly did better than just brute force search. It fails for all but the smallest examples. So that's another one of those things that doesn't really work in practice.
My reading of genetic programming disagrees with you. In fact many have found that mutation doesn't actually help. Pure crossover with a random population seemed to work just fine (for genetic programming.)
The biggest problem is that the search space isn't smooth and even crossover is very mutative on discrete, arbitrary graphs.
