I think one very fundamental fact that you are overlooking, which we pretty much take for granted, is that DNA is the mechanism of inheritance of traits.
What I mean is that my DNA is very similar to my parents' and siblings. It's also similar, but slightly less so, to my grandparents, my cousins, aunts and uncles, etc. The further you go from my known relatives, the less similar our DNA becomes.
This is also true for all living organisms. For everything from bacteria, to plants, to flies, we can look at their DNA, and that of their relatives, and the more closely related (in the familial sense, as in parents, siblings, etc., not in the evolutionary sense, although I'll speak to that in a moment), the more similar the DNA.
So that is demonstrably true for every living thing we can observe today. But what about the past? Well, if this theory of DNA and familial relationship is true, then we can predict that, for instance, my DNA is more similar to the DNA of present-day people in the countries where my ancestors came from than from other countries, even though we have no historical/genealogical evidence of relationship. And this bears out-- a commercial DNA ancestry test shows I share genes with the general populations of the countries my recent ancestors are documented to have immigrated from.
This idea that familial relationship are biological relationship should seem at once obvious but also surprisingly insightful. If I am related to my human cousins because of the mechanism of DNA inheritance and the sharing of a common ancestor (our common grandparents), then it also stands to reason that chimpanzees are literally our familial cousins, very distant, from a common ancestor a long long time ago. Biological relatedness is in fact the same thing as familial relatedness.
If you don't buy that, then you have to explain why it's true that all observable organisms today share familial relatedness through DNA, but for some reason, this isn't true going back into the far past. What changed in the past? When did this start happening?
The only reason we get tripped up in this line of reasoning is because chimpanzees look very different from humans, and we cannot mate. In the past, people thought this was because different species were fundamentally different types of beings. But biological insights, specifically the understanding of DNA, shows that we are all fundamentally the same type of creature, with reproductive incompatibilities developing over time. The tree of life is literally a family tree.
Also, from biochemical understanding, we know generally how long mutations take to arise. Combined with fossil evidence, we can create a pretty good model of how far back divergences arose.