Evidence for common ancestry is not evidence for any specific mechanism. This point I made clear from the get go. Sure there is good evidence for common ancestry. How did one animal get to the next though, assuming that to be true?
That is where inductive reasoning (and parsimony) come into play.
In Darwin's day, all Darwin could do was point to the evidence for common ancestry. He didn't know the mechanism that accounted for it.
We can point to a potential mechanism that has a natural explanation, namely genetic mutation with selection and change in allele frequency over time. There is a ton of evidence demonstrating genetic changes that result in marked changes in body form and function. None of us doubt that, you understand the process, so there is no reason to belabor the point.
What I am hearing you say is that you see a conceptual gap between the types of mutations we see as responsible for sickle cell anemia and the much bigger changes we see in organisms from different species. That is why I tried to point you to the single gene change that appears to be responsible for taming foxes (and presumably dogs from wolves). I did not mean that as an example of natural selection, but the actual mechanism for selection is not important -- for this process it is the change in genetic material and the fact that a selection pressure was applied that is important. We can see fairly big changes in the animals that are tamed and that process is not cultural, it is biological. It depends on a single gene change.
With eukaryotes, and especially complex multicellular organisms, single gene changes can have much more profound effects on the ultimate structure of an organism because those organisms go through long developmental phases. Our DNA is set up in a very complex way with regulatory regions that can alter cascades of genes; but the more important issue is what happens during early development where small changes can result in profound alterations in body planning. That sort of thing does not occur in prokaryotes because they are set up differently.
We have seen speciation events in several different organisms, again where speciation is defined as the development of a strain that can no longer mate with the original strain. Why is this important? Because if one group of organisms can no longer mate with another, they can no longer share genes. We know that with the sorting, including crossing over, in sexual species that new genetic combinations arise, but if two groups of initially similar organisms are separated by geography or inability to mate successfully then they will tend to diverge over time. New mutations occur, new sorting takes place in different ways based on the new mutations; new body plans can emerge with alterations in regulatory regions or in the genes responsible for early development.
That is the thinking behind the development of new species from a common ancestor, how it can occur on a genetic basis. Because of the time involved we cannot observe it directly; we can only recreate what seems to have occurred in the past. The best place to look for this information, what we can use to think inductively about the process, is in the field of cladistics with special emphasis on the molecular data.
Parsimony comes into play because other mechanisms are possible -- it is possible that there is a designer who makes whatever genetic manipulations are necessary to account for new species, for descent from a common ancestor. But such a designer would necessarily require an origin. That origin must be accounted for on natural grounds, which puts us back into the area of evolution by means of natural selection, or supernatural grounds. Whichever route you might choose, that explanation requires more entities than are required to produce the changes we see.
Do we have perfect evidence? Well, of course not. We do, however, have remarkably overwhelming evidence for descent from a common ancestor and the genetic mechanism provides a natural explanation for how changes arise in the first place. To make sense of all this, though, I think you might want to spend more time thinking about embryology and less about sickle cell anemia.