Okay Hooby I've been giving this some thought today to try to work out what your objection is. I think it is simple - you don't like the metaphor of a 'tree' and you are confusing the metaphor with the reality.
Let me put it this way. Forget the tree and nothing is lost from the OP. Imagine that Darwin had never lived and nobody had every thought about evolution. Imagine there were no fossils and nobody had observed homologies or comparative embryology or biogeography or any of the other many many lines of evidence.
Imagine that scientists had simply been studying ubiquitous proteins like Cytochrome C to learn more about their structure and function.
They found that every cell in every living thing from the yeast used to make your beer to elephants all rely on this protein. They find that in every case it is structurally identical - so identical that you can swap them around from any species to any other species and it works perfectly.
They do the sums and work out that there are more ways to create this specific shape than there are stars in the known universe, so they look in detail at the DNA code of this protein in various species. They discover something amazing. The code for cytochrome C in humans is almost identical to that in chimps, a little less similar in apes and a little less similar in new world monkeys. The same pattern is seen in every species and yet all of them work interchangeably.
Comparative genetics DOES NOT depend on an assumption of common ancestry it proves common ancestry. It is the smoking gun. It is the same sort of technique used to prove paternity in legal battles.
Now we can compare the relationships previously discovered using fossils and comparative anatomy with the evidence from DNA and use that to confirm or improve what was known previously.
I hope that helps. If you have any questions feel free to ask but don't leave it ten months and then ignore this answer.