Is 'Neanderthal Thinking' Hate Speech?

If it's impermissible to refer to a group of people in an insulting way--including as a way to insult a different group of people--then it would seem to remain wrong even if the former group of people dies off. And especially if we had a hand in killing them off. So it would seem that it's impermissible to use 'Neanderthal' as an insult--including, e.g., in the phrase 'Neanderthal thinking.' Even against Republicans...

   Not to mention the fact that many of us have Neanderthal DNA. Weirdly, by progressive rules, that's worse than if we ALL had Neanderthal DNA.

   Joe Biden: speciesist.

   And, when you think about it, isn't speciesism even worse than racism?

   There's no sense going just halfway down the rabbit hole...