"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 t

"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.