Why is it that a design hypothesis does not predict this observed nested hierarchy of living organisms?
Of course a designer could create living organisms in a nested hierarchy. But nobody has ever provided a reason it would have to, so it doesn’t predict that. (For scientific purposes, “that’s what the designer wanted to do” is not a reason, because it would apply no matter what the situation is).
As an example, animals can be divided into those with hair and milk (H/M), and those without hair and milk (NH/NM). A designer who can make H/M and NH/NM animals could almost certainly make NH/M and H/NM animals. But there aren’t any. And there aren’t any other situations where there are significant differences between the groupings as determined by various characteristics.
Moving on to the chart:
A node is a spot in the diagram where two lines meet.
Each node and each group name on the right represents a group of organisms (I’ll call it a population) at a moment in time.
Each line segment from node-to-node or node-to-group name represents the progression of a population through time (a lineage). As time passes, individual organisms within the population die and new individual organisms are born.
Does common descent have to result in a nested hierarchy? Of what?
Let’s take a look at the mangabey and the baboon. Follow the line segments back from each of those populations to the node where they meet, and call that population MB.
The population MB at some point split into two populations, M and B. The current populations M and B do not overlap with each other (no organism is both a baboon and a mangabey) and also do not overlap with any other population (no mangabey or baboon is a member of any of the other groups: human, chimpanzee, etc.).
Together M and B make up the entire set of all living descendants of the MB population that are included in this chart. There may be others, but we are ignoring them for now.
Bingo, you have a nested hierarchy, of populations and their descendant populations: populations B and M are nested in population MB and its descendant populations.
Moving left from the MB population, the next node is MBR, where the lineage that leads to living rhesus monkeys (R) broke off. MB and R are descendent populations nested in population MBR and its descendant populations.
None of the other populations are in the group of populations and their descendent populations that we call MBR. And any other group of populations that has MB in it also has R in it.
So yes, common descent with imperfect reproduction can only result in a nested hierarchy of populations and their descendant populations. (With minor exceptions, of course, like hybridization. Imagine there are mangabeys and baboons that are separate populations. If they interbreed, their hybrid descendants will belong to both the M population and all its descendant populations and to the B population and all its descendant populations.)
So why do we think there is common descent?
Because living organisms fit into a nested hierarchy.
Because living organisms and fossils fit into a nested hierarchy.
Because no matter what characteristics you consider, the hierarchies are essentially identical to each other.
Because physical characteristics are correlated with genetic characteristics.
Because genetic characteristics are passed down from generation to generation, imperfectly.
Now, we may not have enough evidence to determine the place in a particular hierarchy where a particular organisms or fossils belong, and there are known factors that can lead the evidence to be in conflict. (incomplete lineage sorting).
But the big picture is pretty convincing.