I think it’s more likely that Neanderthals died out because their crazy muscular bodies demanded a lot of calories — more than 50% more than a modern human does. So in times of famine or bad weather, they couldn’t find enough food to keep those muscular bodies fueled. Also, while they could sprint faster, we’re better at long distance running. So we excelled at hunting in non-forested areas.
I don’t disagree at all.
I would also note that a neaderthal female would be bigger an tougher than all but the biggest and toughest homo sapien sapien, so any gene sharing was almost surely the result of neaderthal men breeding with HSS females -— although some men still like to get tossed around.