Was this the form of intellectual superiority that Homo sapiens flaunted when they were among “dumber” apes?
So "our predecessors" isn't specific enough. One might mean our pre-ape ancestors, like you do, or the subgroup of apes that is hominids.
I mean a new "idea" is far from being able debunk anything, right?
Also, the debunking likely refers to the really old progressivist idea that any species that went extinct must have done so because it was inherently "inferior", instead of less well adapted to whatever changes the environment threw at them.
Either way, from what I understand the most recent well-supported theories of what caused neanderthal extinction is that their basal metabolic rate was so much higher compared to ours (due to adaptations to the ice age), that this simple caloric difference gave their body plan a long-term disadvantage over ours after the ice age ended[0].
And when I say "body plan" I refer to the part where apparently early humans and Neanderthals interbred so often that about 20% of Neanderthal DNA is still present in the modern human gene pool, and most humans have about 1 to 4% neanderthal DNA[1]. Similarly DNA from homo sapiens has also been found in neanderthal remains[2].
So were kind of reaching this point where Neanderthals and Denovisans seem to not quite match the definition that most people use when they say "different species". Our ancestors interbred a lot, and the genes that were advantageous stuck around. At that point using the word "extinction" without caveats becomes a bit misleading I think.
[0] https://www.youtube.com/watch?v=a6d3uHSFxkM