Because denying a group of people (say, based on ethnicity or skin color) the right to exist is equivalent to hating them.

Take a person who believes that "there needs to be a country for white people and white people are innately better able to form productive societies". This person would clearly be a white nationalist and a white supremacist, right?

But they may also have no hate towards other ethnicities or desire their deaths. If pressed, they might even say that their vision of a "pure" society isn't worth the deaths of minorities that would come about if they tried to implement it.

I think too often we confuse the stereotypical example with the definition. The stereotypical white supremacist hates minorities, but the definition itself doesn't require it (I know of no surveys that would tell us what proportion of white supremacists match the stereotype).

You can't be a white supremacist without thinking other races are inferior. That's hateful by definition. They are stereotyping an entire race in a negative manner. They are denying the basic humanity of billions for what end? The Third Reich didn't immediately start throwing Jewish people into death chambers. They had to build up to that moment by dehumanizing their victims.

> That's hateful by definition.

My whole point is that you (and many others) are using a new definition of "hate" which doesn't match the old one. "Hate" used to be an emotion, a feeling, a dislike of something and a wish to see it destroyed.

One can feel superior to something without having any dislike of it or a wish to see it destroyed. I consider myself superior in many respects to the rocks in my back garden, but I neither dislike them nor wish them destroyed.

A supremacist may consider themselves smarter or prettier or taller than some other group, but that does not necessarily mean they want to destroy the other group.

I kind of get your point (i think?) but maybe you shouldn't try to belittle the use of the word "supremacists" in the context of modern language. Try looking up a definition if you are unsure. Maybe you disagree on the definition but that is probably the mainstream one...

At your prompting I've tried to look up a definition, but there doesn't seem to be a commonly accepted one... merely usage where the scope of the terms varies from one source to another.

That is simple.

Any left-wing should be allowed, any right-wing stuff should be denied.

Few exceptions exist on the western side, Spain is probably the most remarkable case. Reason why you wouldn't often hear much about what happens there, unless it is something negative to bash the right-wing people there.