“Interbreeding with fertile offspring says there was only one species involved unless”
Yes, exactly. If they want to say that Neanderthals were a separate species, then they need to change the definition of “species” to remove the fertility/sterile elements. Maybe it’d be more accurate to say that Neanderthals were a “race.”
In flowers they would be varieties.
I don’t think they died off so much as interbred until there was only one population. Archaeologists and anthropologists say that the Neanderthal population never exceeded 100,000, and if the two groups mingled for 7000 years then the smaller group would simply have melted into the far larger group and woould leave traces of DNA in the descendants, like maybe the 4% or so that is reported.