First let me say that I do understand your “technical” premise about using the terminology “different species” when referring to neanderthals and Modern Humans. But there is something behind that when considering interfamilial species of the same genus. In this sense I think they are referring to hybrid crosses similar to what you get when your cross a Coyote with a domestic dog. While they are still both of the Canine genus, they are indeed different species of canines. Which by the way are viable and fertile and can produce offspring, just as the Beefalo hybrid from the Bison Cattle cross is fertile can reproduce as it’s own cross species. So while Bison and Cattle are of the same ‘Genus” Bos, they are still considered different species.
I think part of the Mule issue is that we have never given them that rare chance to possibly end up with fertility or viability, we castrate them all as a rule and practice. As for females, there are several cases where a female Mule has indeed bore offspring. But when it comes to even cross genus species hybrids, there are more known instances than I think you realize... So I think the possibility under the right circumstances of survival threat could change these things. But I’m not at all an expert just going by what I have read.
https://duckduckgo.com/?q=Cross+species+hybids&t=lm&ia=web