The highlighted statement is not correct. There is no requirement that a new species' ancestor species go extinct.
We each have a brain to think for ourselves. Most people, for example, aren't aware that after nearly 50 years of intensive efforts, science abandoned laboratory efforts in the late 90's to create the simplest precursors to a living organism.
The TOE does not cover where life came from.
I am sure, from a strict viewpoint, that you are correct that the theory of evolution doesn't cover the origin of life-- however I am also sure that you know as well as I do that the way in which it is taught in school and presented in popular sources results in the vast majority of people believing that it does. This result is not accidental.
Personally, I don't see any reason why the theory of evolution has to stand in opposition to the existence of God, but even so, I don't believe in it in any but a micro way. I don't think the evidence supports it, and the rabidity of belief among supporters is so insistent, and so intent on calling the other side stupid that I think the hubris will catch up with them. Sooner or later, I think the zeitgeist will shift, more evidence pointing the other direction will be found, and the theory will be utterly discredited. But, we'll see. ;)