Nazism comes from National Socialism first and foremost.

This is wrong for different reasons. “National Socialism,” as a political idealogy, played virtually no role in Nazism, aside from the use of a politically trendy buzzword to garner public support in the early years of the party.

From Wikipedia

In 1920, the DAP renamed itself to the Nationalsozialistische Deutsche Arbeiterpartei – NSDAP (National Socialist German Workers’ Party, commonly known as the Nazi Party). Hitler chose this name to win over left-wing German workers. Despite the NSDAP being a right-wing party, it had many anti-capitalist and anti-bourgeois elements. Hitler later initiated a purge of these elements and reaffirmed the Nazi Party’s pro-business stance.

and the ideology itself was not inherently antisemite

The Nazi Party and Hitler are (like Nazism and antisemitism, funnily enough) inextricable. In every practical sense, the Nazi Party existed solely as a tool for Hitler to use. Antisemitism was the foundation of Hitler’s political career. Again, it’s impossible to say that “the idealogy of Nazism was not antisemitic” when it was led unilaterally by “Antisemite: The Guy.”

but remember that the Nazis also targeted other groups and ethnicities as well.

The majority of these other people were political enemies. Communists and Westerners did not face even a fraction of the social hate or dehumanization as the Jews. Yes, it’s wrong to say that the Jews were the only target of the Nazis, but it’s also wrong to say that they were not the target of the Nazis.

But dehumanising groups of people and referring to them as vermin (to make their idiot followers easier to persuade that the groups are not real people) is definitely straight out of the Nazi playbook.

Not “groups of people,” this is what I’ve been saying the whole time. Dehumanizing Jews and referring to them as vermin is straight out of the Nazi playbook. Many people have referred to target groups as vermin throughout history. Julius Caesar did it to the Gauls. Andrew Jackson did it to the Amerinds. Mao did it to the capitalists. None of those figures are called Nazis because antisemitism is the cornerstone of Nazism and Nazism without antisemitism is an oxymoron.