Modern humans having some admixture from them doesn't contradict the idea that the ice age was the bigger problem than direct competition.

My own guess would be the final years of the Ice Age saw too many changes in the northern hemisphere, erratic weather that wiped out everyone, but I have not seen that in an official source, except in discussions of Neanderthals.

Modern Humans can be an admixture of several species if the ancestors to modern humans either migrated south to Africa or continued to survive in some place like Arabia, but perhaps we just haven't found the right fossil yet. Assuming Sapiens and Neanderthals mixed together in the Levant then they either survived in the Levant or they moved south into Africa and then later expanded out of Africa.