If you are going to cite research, please actually cite it. Don't just vaguely assert "Scientist X sez" and make me do all the legwork. I couldn't find the peer-reviewed paper where Shapiro claimed that mutations are non-random; what I did find was numerous people pointing out that he's kind of a crank. The idea that mutations are fundamentally non-random is unsupported and makes no sense given the paradigm of how mutations are formed, and there are numerous experiments (such as bacteria gaining antibiotic resistance genes through mutation despite no exposure to that antibiotic) that show just how random it is.
But even if the mutations weren't completely random, it should still form a nested hierarchy fairly consistently. And hey - the methods for this are all really easy. There's a ton of cladogram programs online (the video cited one, and others are a quick google search away), and you can play around with making X% of the mutations non-random to see where the threshold is. Would be an interesting (if rather pointless) experiment. I strongly encourage you to actually do the legwork to back up your claims, rather than to just throw them out apropos of nothing.
The explanation is that descent with modification necessarily leads to a single natural cladogram which is the most parsimonious that will line up with the descent. We have the mathematical models to prove this, as detailed in the video. This is not some assumption, this is a proven mathematical theorem.