I think they never were a distinct species. How can you tell something is "neanderthal DNA", if the vast majority of genes is shared anyway? So let's say we have three common alleles of a particular gene: GATTACA, GATCACA, and, GATAACA, then you find a neanderthal homozygous for GATAACA, and you somehow decide that all its occurence in modern people are from "neanderthal admixture"? Why?
You can see people looking more or less "neanderthal" in different times and places, even within the short period of photographic evidence. The difference is environmental.
"Species" is an artificial categorization created by humans to make it easier to talk about population groups, but as you note, the real world is more complex than that, so the lines are indeed blurry since both Neanderthals and homo sapiens share a relatively recent common ancestor and could apparently interbreed.
But over the decades of research into this field, scientists have had exactly this discussion about where to draw the line between "species" of Homo and have various standards for making this distinction, though certainly some of them disagree with the general consensus, and things change over time.
But there's plenty of genetic evidence and research that have gone into these findings, and plenty of Neanderthal genes that don't appear in modern humans. Just because it goes against your gut intuition doesn't mean they are wrong to make this distinction. There are decades of literature on the topic if you are actually interested in learning about it.
I thought species meant collection of individuals who could breed with one another.
So foxes and dogs/wolves are distinct because, while clearly very related, their chromosomes are factored differently (haploid numbers), whereas wolves and dogs and coyotes are all compatible, for examples.
Mitochondrial DNA is one way they could tell neanderthals were a distinct species. Teeth shape as well. It's not just looking at the DNA itself, but the behavior as well, such as isolated reproduction, different adaptations, and ancestors as well. Modern humans can be traced back to a particular ancestor, and Neanderthals can be traced back to the same ancestor, where they branched off from. Neanderthals diverged from homo heidelbergensis around half a million years ago, and modern humans did as well, though later. It's not just about looks.
Regarding your particular point about identifying "neanderthal DNA", consider that most genes are made up of thousands of base pairs. Even though there might be individual variation, it is trivial to distinguish between, say, chimpanzee and human versions of certain genes based on what we know about the populations. We also know the rate of evolution of genes and how they diverge, which means we can be pretty sure that a given version was acquired through interbreeding versus spontaneous mutation.
Nobody claims that we interbred with chimps though. And there is the recent historic evidence.
Native Americans look very neandertalish in old photos, and even older drawings suggest that they used to look even more so. The difference is so stark that they were believed to have been pretty much exterminated, until DNA showed that they probably joined the Spanish, and their descendants now call themselves "Mexicans"... When they adopted European lifestyle, they started looking almost white.
Or the Chinese. Slanted eyes used to be seen as the defining feature of Chinese people. Almost no modern Chinese have slanted eyes today. It can still be seen occasionally in people who came from some isolated place in Western China, or something like that, but it seems absurd that something so rare could make it into a stereotype.
Too much weight is put on genes. I think it's somebody's nice fantasy that we solve all our problems through gene editing, but it isn't so.
The movie name is formed from DNA letters on purpose.
Anyway, the main difference that distingueshes sapiens is the poorly grown face, and there is no known reason why people should evolve an ungrown face.
What do you mean by poorly grown? A robust neanderthal skull can handle injuries from megafauna better. Our better strategy (potentially through more developed language), finer weapons, and slightly better ballistic aim combined to make us the better hunter. The evolutionary pressure for robust facial structure was not there - we evolved in a hot environment where a more lithe body was better adapted to long-distance persistence hunting.
Your choice of "poorly grown" reveals that your mental model of evolution and development is poorly grown. In what way is it "poorly grown" or "ungrown" except by your standards of what a "human" should look like, which is only based on what modern humans _do_ look like?
You can see people looking more or less "neanderthal" in different times and places, even within the short period of photographic evidence. The difference is environmental.