If only two of your six parameters actually affect the result, a grid wastes most of its trials varying the irrelevant four at fixed values of the important two. Random sampling varies everything at once, so 60 random draws typically explore the important dimensions better than a 64-cell grid.
It is also easy to budget: decide you will run 500 trials, then stop. That fixed trial count is worth recording, because it is exactly the number you need for any multiple-testing adjustment later.
Random search does not solve overfitting. It finds a good cell faster, which means it finds a lucky cell faster too.
Related: grid-search, bayesian-optimisation, parameter-optimisation, multiple-testing