I don't know if I can make a syllogism out of this. Maybe you can after you read it.
I'm sure neither of us can.
The starting point is your definition of significance:
Value, importance, in the Great Scheme of Things, not just to ourselves or selected others in our lives.
That's not anything that science addresses at all. No science has anything to say about significance.
Since science became God as it were, we are told we are arrogant for thinking highly of humanity as such.
You are misunderstanding that (and some scientists also misunderstand it). When studying homo sapiens, science takes a value-neutral stance toward them, as it does for anything else it studies. But this is just a stance. It carries no weight on how society should value humans.
Scientists also follow pretty strict ethical rules, when using people as experimental subjects. That should at least hint that the value-neutrality is merely a stance, and not how people are really viewed.
Evolution treats us as nothing special at all, just another animal, ...
But this is not just evolution. Biology treats us as a collection of cells. Astronomy treats earth as an insignificant planet orbiting an insignificant star. Again, this is a stance scientists take, so that there method of study will be uniform and consistent.
If scientists didn't consider us significant, they wouldn't be so diligent about doing their science.
This is something we must accept by accepting evolution, to tie it into the title of the thread.
But we only have to accept it as a stance scientists adopt while carrying out their scientific study.
Most physicians will not handle medical problems for their family (other than trivial ones). And it is for the same reason. In order to do the best medicine, one must take a neutral stance toward patients, which is very hard to do if the patient is family.
Sure, Dawkins treats the stance as if it is the reality. But that comes from his atheism, not from his being an evolutionist. We don't have to follow Dawkins in that respect. It is entirely possible to maintain one's Christian values with respect to the significance of humans in the great scheme of things, and still be an evolutionist. Many people indeed do that.