Re: Projection much?
Based on genetic research into this it seems that all that humans got from Neanderthals was wast amount of problems and flaws. Those issues might suggests a wider problem (not yet discovered?) and might be the reason for their extinction. The human race got few good genetics from this mixing. I am not sure if it is something that would have happened on its own with time and evolution.
Science article, https://www.sciencedaily.com/releases/2014/01/140129134956.htm (2014)
News article, https://www.smithsonianmag.com/smart-news/new-studies-show-neanderthals-gave-us-some-good-genes-and-nasty-diseases-180960870/ (2016)