Given our recent performance in extinction level interspecies violence and our considerable capability of intraspecies violence I find your characterization of violence as only one of many competing theories surprisingly ... endearing. But you are not wrong, given the low populations involved on both sides it's not impossible at all that violence wasn't the decisive factor.
That isn't evidence that we killed the Neanderthals for food. Rather than, say, having added some ecosystem stress through competition that left them less able to handle a major volcano 40,000 years ago.
I think that if human practiced widespread and consistent cannibalism, then it stands that we likely ate Neanderthals and other homo species to some degree