“the random walk of evolution “
Consider that evolution may have (and probably did) evolve such that rapid changes, more variability, and evolution driving evolution was and is better.
That is rapid evolution produces a more fit species than a slower evolving one. So evolution may not be a random walk and is a process driving itself.
You can prove that sexual reproduction is an improvement for species survivability over asexual reproduction. This is a case where evolution “programmed itself” for faster genetic changes and is not merely a random walk.
And for the next to last statement. You can prove that sexual reproduction is an improvement for species survivability over asexual reproduction. Obviously false or have you forgotten about bacteria?