The evidence is not definitive, but what evidence there is points toward no interbreeding. There is certainly enough evidence to declare that the headline of this article is false. By all appearances, we did not sleep with Neanderthals (even if we did). It is not 'appearances' that makes people such as this lady imagine that we did; it is nothing but merely their fantasies about how Paleolithic hominid populations in Europe 'must' have behaved..
For the sake of this discussion I am going to call just use "Homo Sapien" to denote our species.
The evidence may be a bit misleading depending on how the DNA of the two species (if Neanderthal and Homo Sapiens are two species) interact. For instance the lion and the tiger have at least that half million years of evolution separating them and they can still successfully interbreed (although it doesn't happen in the wild). We could force interbreed Humans and chimps in the lab today and it wouldn't be that much of a "force" with current technology.
The question is did Humans and Neanderthals co-mingle. I think the Answer is probably yes. Humans are very promiscuous and even if it was just rape on one side or the other it most likely happened much more than once in the thousands of years of Human Neanderthal interaction.
Did these co-minglings result in children. I think we can say we don't know. Your assertion that the children must have grown up in Human communities is a bit misleading. If mixed children were being born into Neanderthal society they probably developed at least some Homo Sapien cognitive ability and so the Neanderthal tribe would quickly begin to leave cultural artifacts which resemble Homo Spain artifacts which did in fact happen when the two cultures lived near each other. It is possible that Neanderthal tribes died out not as a result of increased competition but as a result that their children quickly stopped being Neanderthals if 1/2 breed children were growing up in their midst and also seeking mates from nearby Homo Sapien populations. In fact the change over could be so rapid to leave very little in the way of fossil remains of the population that is interbred. (In fact some bones that have been found suggest half breeds did exist but they are still debatable)
Nothing can be said for sure at this point and you seem very certain and that is just not good science with the available data.