It might be helpful to discuss a specific case where scientists thought they may have found evidence of non-random mutations.
What happened is they knocked out parts of a lactase gene responsible for lactose metabolism. They starved these bacteria in the presence of lactose, and in response they saw an increase in the number of mutants who recovered lactose activity. On the face of it, it appeared that the presence of lactose increased the rate at which mutations recovered lactase activity. If the presence of lactose directly increases the rate at which the bacteria specifically recovered lactase activity then those would be mutations that are non-random with respect to fitness.
However, the story took a different turn. As it turns out, when E. coli are starved it causes DNA damage. The bacteria respond to DNA damage by turning on genes that increase recombination and mutations, which is called the SOS response. These mechanisms didn’t specifically increase the mutations in the lactase gene, they increased the rate of mutations throughout the entire genome. Therefore, it was finally concluded that these mutations were random with respect to fitness afterall.
This is why I say that Shapiro is misleading people. Increases in the rate of random mutations is not an example of non-random mutations, and yet he tries to act as if it is.