Perhaps you are not aware of research over the past decade or so?
First of all, there’s no doubt at all that Neanderthals and biologically modern humans existed at the same times.
Indeed, if we use the usual date given for the first biologically modern humans, 200,000 years ago, then Neanderthals existed for the first 160,000+ years.
Obviously, not together at the same times & places until around 45,000 years ago, for maybe 10,000 years or so.
That’s when events got interesting, and nobody knows for sure what happened, except that when it was all over, Neanderthals were gone, and all non - Africans share up to 4% “Neanderthal DNA”.
So figuring out how & why that happened is a great quest of current research.
Perhaps you are not aware of research over the past decade or so?
There is not anywhere near 100% agreement on this. There are some interesting outcomes of DNA research. Chimpanzees have 95% of human DNA but a Carrot has 80% of human DNA.