We shouldn't use the word "species" lightly with hominins. There's no accepted way to properly classify archaic humans.
For example, some paleoanthropologists classify most archaic humans as h. sapiens. Anatomically modern humans become h. s. sapiens, neanderthal become h. s. neanderthalensis. This incorporates the middle pleistocene hominins from mainland Asia pretty well to boot. Many of those same people also use the "conventional" binomial terminology when they're not making a very specific point, so you can't just look at usages to understand where they're coming from.
There's also a hundred other classifications, some giving neanderthals their own species, others including it with heidelbergensis, and so on. None of them has clearly "won" and probably won't while we keep publishing "new" transitional forms every couple of years.
Sure... and it would help if there was more consistency in general.
But given that there isn't, we should at least maintain internal consistency. That's what I was commenting on above. You can't use one conceptual framework for Neanderthals/Eurasia and another for Sapiens/Africa. It's confusing... and you end up with statements that are essentially false.
Neanderthal and Sapiens exist approximately concurrently. The "classic neanderthal" form doesn't predate sapiens. It only predates Sapiens if we define "Neanderthal" from the point of divergence. If we do that, we need to define Sapiens the same way. It can't be the establishment of a founder population for one and the emergence of a distinct form for the other.
I'm not advocating for one system or another... or even for the establishment of one consistent system necessarily.