The part of the randomness that I struggle with is the sheer volume of data to be manipulated into useful output (a form of the infinite monkeys with typewriters adage). For example, the smallest genome I can find with a quick google is Carsonella ruddii, which has approximately 160,000 base pairs in its genome.

That many base pairs represents 4^160000 possible combinations. Brute forcing the genome to something better is clearly possible, but if you converted all of the matter in the visible universe into DNA and mutated every strand a trillion times a second for a trillion years, it's very possible you would never even encounter a viable strand of DNA for that organism.

How many reproductive events have occurred in the lifetime of this planet and how has that led to the vast genetic diversity that we see? You can upper bound it by taking some multiple of the biomass on earth times some percentage of that which is DNA by some average reproductive cycle duration, and you're going to find a shockingly paltry number in the face of what exists today.

edit - Look, wikipedia already did the work for us (no idea of the methodology): https://en.wikipedia.org/wiki/Biomass_(ecology)

>The total number of DNA base pairs on Earth, as a possible approximation of global biodiversity, is estimated at (5.3±3.6)×10^37, and weighs 50 billion tonnes.

So to box it in a bit, 5e37 base pairs is 5e37/160,000 or (rounded up) 4e32 Carsonella ruddii. If they reproduce every 4 minutes for 4.5 billion years you have ~6e14 generations for a total of 3e47 distinct instances of DNA as a terribly rough approximation for an upper bound of genetically distinct individuals.

That's almost enough to fully explore a section of DNA that's 80 base pairs long. Most genes are ~1000 base pairs. A butterfly has 100,000,000 base pairs in their DNA. Humans have ~3,000,000,000.

Randomness is clearly a factor, but it seems insufficient.