I’ll try a more detailed reply to @Perry_Marshall here.
I did look at your blog post, and my reaction is similar to that of @swamidass, but lets avoid repeating that and get to the question of randomness. In ordinary life, “random” is used with a range of different meanings. For biology, particularly evolutionary theory, the main concern is that mutations are random with respect to fitness. And that mostly means that mutations do not seem to be purposely oriented to improve fitness. If we found that mutations were usually beneficial (improved fitness), that would pose a problem for evolutionary theory, even if the mutations satisfied mathematical tests of randomness. So the use of the term “random” in evolutionary theory is mostly a way of saying that mutations do not appear to have any bias toward improved fitness.
If there is a locus on the genome where mutations are almost always fatal or seriously detrimental, then evolutionary theory already predicts that the mutation rate should be lower at such a locus. So variation of mutation rate is not consider non-random.
The traditional account of evolution says that mutations are copying errors. And it looks to me as if you are really arguing about that view of mutations. But random need not imply error. If I’m making a cake, I will stir up the cake batter. Stirring it up is a form of randomization, but it isn’t accidental or an error. Random is not the same as erroneous or accidental. I’m suspecting that you (and Shapiro and others) are making a mistake by arguing against randomness of mutations. If you were instead arguing that mutations are not accidents, but are instead part of how the system works, then I would probably be agreeing with you.