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
A male horse and a female donkey have a hinny. A female horse and a male donkey have a mule.
But hinnies and mules can't have babies of their own. They are sterile because they can't make sperm or eggs.
They have trouble making sperm or eggs because their chromosomes don't match up well. And, to a lesser extent, because of their chromosome number.
A mule gets 32 horse chromosomes from mom and 31 donkey chromosomes from dad for a total of 63 chromosomes. (A horse has 64 chromosomes and a donkey has 62).
Chimeras, Mosaics, and Other Fun Stuff | June 20, 2007 | Monica Rodriguez
Once in a while there are offspring (poorly documented, anecdotal evidence, and probably not yet genetically tested) but the offspring will be either a donkey (with a donkey mother) or a horse (with a horse mother), rather than a hybrid offspring. Similarly, the chimp (and bonobo I suppose), gibbon, gorilla, and orangutan (and perhaps the cryptoanimal the orang-pendek) have 24 chromosome pairs; humans have 23 chromosome pairs. Reportedly/supposedly there have been attempts by some infamous dictators to create a chimp-human hybrid and then mass-produce them for use in a fast-maturing hybrid ape-man army. Bupkis.