Just will add this: we don't know, maybe the Neanderthals already could speak thousands of years before the Home Sapiens learned to do so.
Then the original Neanderthal civilization had a sharp edge over the Sapiens, through thousand of years.
Ok, everyone says here "but technological development SHOULD have happened". And I would say "not necessarily", and it actually could happened a couple of times, but it didn't had the written word advantage to keep building upon learned stuff (you keep discovering-then-forgetting the same things over and over), or maybe there was other factor limiting the Neanderthal technological development (neurological, strong "bad-to-tech" cultural practices, etc.).
You say "not probably", but Homo Sapiens just recently (re)learned to wash hands as a safe method to better health outcomes, maybe for the 40th time in several thousands of years across a dozen of civilizations / tribes who already know about washed-hands = good health:
Maybe those original Home Sapiens migrants just went to Neanderthal territory, maybe established "homo sapiens republics" (different culture, technological practices), and eventually got absorbed into the lasting main Neanderthal culture.
Just like it happens to migrants in our time, in just a few years, not thousands.
It could be that after 600.000 years, Neanderthals got somehow behind minimal species fitness required to endure lasting species survival, and they were - this time - absorbed by the Homo Sapiens.
I don't think there weren't ancient / primitive civilizations or tribes, it's just that they were fully erased from history as they didn't left any written history or well constructed monuments and/or big cities that we could find after 100-150.000 years (no need of "great events" to erase cities, just build the houses with mud bricks, wait 30-40k years).
At looking for clues about why those initial waves went exting, maybe we could take a look at how more recent tribes or even old civilizations like Incas, Mayas, etc. made it through a couple of centuries, and eventually got "extint".
Some of those groups were surrounded by other tribes, and tried to expand their territories, with different outcomes.
Some didn't tried to expand, they just got well adjusted to their original territory and didn't need more resources.
But everyone in those groups eventually disappeared from history.
Even when the american continent was discovered, the remaining "true" mayas didn't remembered or indicated that those big monuments, cities ever existed already taken / hidden by the jungle. Many of those also, were just found literally hundreds of years after having seen the last true mayas.
Then the original Neanderthal civilization had a sharp edge over the Sapiens, through thousand of years.
Ok, everyone says here "but technological development SHOULD have happened". And I would say "not necessarily", and it actually could happened a couple of times, but it didn't had the written word advantage to keep building upon learned stuff (you keep discovering-then-forgetting the same things over and over), or maybe there was other factor limiting the Neanderthal technological development (neurological, strong "bad-to-tech" cultural practices, etc.).
You say "not probably", but Homo Sapiens just recently (re)learned to wash hands as a safe method to better health outcomes, maybe for the 40th time in several thousands of years across a dozen of civilizations / tribes who already know about washed-hands = good health:
https://es.wikipedia.org/wiki/Esenios