My guess is that the fact that no organisms exist with a Neanderthal genome defines them as extinct. Where one draws the line is more art than science I guess ... I know that there are some genetics in us (like the HMG group of proteins) that are ancient, but work so well that we still retain them. That doesn't mean the first species to have evolved them isn't extinct, it just means we evolved from them.
Well, I don't think that quite matches the scientific concept of "species". By your definition, almost all species who were alive 50,000 years ago would be considered extinct, but hardly any biologists would agree with that. It's true that no humans alive today have 100% Neanderthal genes, but it's also nearly certain that there are no living humans with 100% Cro-Magnon genes, either. What happened would be considered a mixing of several human sub-species after migrations of one or more African groups into Eurasia. The Cro-Magnon sub-species disappeared, too, and modern human Caucasian and Asian sub-species are the results of that mixing. This sort of thing happens in species all the time, when conditions allow such genetic mixing, and the result is rarely considered a new species.
The fact is that modern humans are all one species. We can and do interbreed when groups mingle, and there are no groups of modern humans that are genetically incompatible. If sub-species "disappear" by genetic mixing, that is usually not called an extinction event. It's just the routine and normal mingling of subspecies.
An interesting contrast is that most North American duck species are known to hybridize occasionally, and the offspring are usually fertile. Does this mean they're really all one species? No, because they all mingle a lot, but interbreeding is rare. They have "behavioral" species-separation features, mostly based on female mate choice. The females are mostly all mottled brown (protective coloring), and the males often approach females of other species (because they can't tell them apart either ;-). But the females usually only accept males that have the "right" color markings; the others are ugly to them. This suffices to keep the species separate, though there is probably a very low level of genetic interchange between many of the species.
But humans aren't like this. Even if we do generally prefer mates in our own subspecies, most of us do find many members of other subspecies physically attractive, and we'll mate with them given the opportunity. This means that we really are all the same species. We now have good evidence that the Neandertals were merely another subspecies, because when they had the opportunity, they did interbreed with those slender, dark-skinned folks who migrated into their territory. They did so often enough to produce a new subspecies that's physically distinct from either of the earlier two (or three or more).