It is really not far-fetched that these passages describe some ancient memories of coexisting with another quasi-human race. And certainly the narrative of interbreeding is tantalizing. Perhaps if it did not actually happen this way, we believe it now, because many Christians would need to interpolte truth about the past and reconcile these passages.
https://bible.usccb.org/bible/genesis/6?1
1 When human beings began to grow numerous on the earth and daughters were born to them,
2 the sons of God saw how beautiful the daughters of human beings were, and so they took for their wives whomever they pleased.
3 ...
4 The Nephilim appeared on earth in those days, as well as later, after the sons of God had intercourse with the daughters of human beings, who bore them sons. They were the heroes of old, the men of renown.
27 They do not rest with the warriors
who fell in ancient times,
who went down to Sheol fully armed.
Their swords were placed under their heads
and their shields laid over their bones;
For there was terror of these warriors
in the land of the living.
33 There [in Canaan] we saw the Nephilim* (the Anakim are from the Nephilim); in our own eyes we seemed like mere grasshoppers, and so we must have seemed to them.”
Neanderthals were fully human. They were the same species (Homo sapiens) as us. We could interbreed and produce viable offspring and we often has similar customs (burial of the dead, lithic art, tool manufacture and use). There was no coexistence with a quasi-human "race", there was just coexistence with different peoples.
As far as I know, the discussion is ongoing and H. neanderthalensis is still a current classification. But yes, the consensus is moving in the direction you describe.
We'll probably never know how they were perceived in society back then, if it was e.g. like Asians versus westerners now.
Neanderthals are classified as Homo neanderthalensis and not Homo sapiens.
Neaderthals are classified as Homo sapiens neanderthalensis, we are classified as Homo sapiens sapiens.
Of course, it depends on who you are reading. There are plenty of disagreements on taxonomy among the published academics and 19th-century literature is still influential, and pop-sci writers and their AI pets rarely do thorough research.
Just last week I was in the Coa valley in the iberian penninsula observing upper paleolithic rock art that could only be credited to Neanderthals (because there is zero evidence sapiens has migrated to Portugal 35,000 years ago; Spain was still covered in ice sheets and provided a physical barrier). Excavations had found stone tools and even a burial at nearby sites. Chances are really really good you wouldn't be able to pick out a Neanderthal if one were wearing modern dress walking down the street. Those academics who classify them as a separate species are merely propagating old imperialistic beliefs from the 19th century.