Adaptive evolution is driven by the theory of natural selection, as discovered by Charles Darwin. A common misconception is that natural selection is random. It is not. It is the opposite of random. It is a deterministic algorithm.I think I understand what he is trying to say, but his concept of randomness is faulty.
There is a legitimate academic controversy over whether evolution is caused by natural selection, or whether a substantial part is caused by random chance.
Darwin defined natural selection as "survival of the fittest". Fitness is defined in terms of survival of progeny. It is primarily a tautology, with Darwin using the term to distinguish from the artificial selection used by farm animal breeders.
Random chance does not cause anything. Saying that something is partially random is just a way of saying that something is partially unexplained. Randomness is not an explanation. There are determinists who do not believe that there is any such thing as true randomness.
Saying that natural selection is non-random is also confusing. He is not saying that we have some algorithm for deciding survival of an animal or species. So what does it mean to be non-random? Usually someone says "non-random" to mean that he has some way to predict outcomes with certainty.
To the extent that natural selection is a tautology, it is hard to see how a tautology can be the main driver or cause of anything.
Dawkins is sympathetic to multiverse theories, and starts one of his books with an argument that most people never get to be born. Again, this is a goofy idea of chance. There are multiverse advocates who would say that any physically possible DNA sequence is realized in some alternate universe. The vast majority of those are never conceived in our universe, of course.
Dawkins has spent most of his life explaining Darwinian evolution to people, and he is world-famous for doing it, so I have to assume that he finds that these explanations are effective.
By comparison, here is how Herbert Spencer describes it in his 1864 biology textbook, a mere 5 years after Darwin's famous book:
This survival of the fittest, which I have here sought to express in mechanical terms, is that which Mr Darwin has called "natural selection, or the preservation of the favoured races in the struggle for life." That there is going on a process of this kind throughout the organic world, Mr Darwin's great work on the Origin of Species has shown to the satisfaction of nearly all naturalists. Indeed, when once enunciated, the truth of his hypothesis is so obvious as scarcely to need proof. Though evidence might be required to show that natural selection accounts for everything ascribed to it, yet no evidence is required to show that natural selection has always been going on, is going on now, and must ever continue to go on. Recognizing this is an a priori certainty, let us contemplate it under its two distinct aspects.Likewise evolution may be defined as genetic changes in a population from one generation to the next, and, once enunciated, the truth of that is so obvious as to not need proof.
Of course Darwin knew nothing about DNA, or even that genes are discrete, or that mammals get half their genes from each parent.
I would think that Dawkins would be better off explaining evolution by skipping the randomness, the Darwin, and the tautology. He could say something like this:
Individual differences are coded in DNA, and are derived from a mixture of parental DNA. When the DNA makes it poorly suited to the environment, it dies out, leaving others that are better suited.