Death metal has its roots in the anarcho-punk movement. Black metal is mainly prominent among neo-Nazis. So yes, people get a little pissed if you conflate the two.
Tossup between that and Nu-Metal, really. One's for suburban white kids, the other is for Norwegian mall pandas and suburban white kids who wish they were Norwegian mall pandas on their way to bvrn a chvrch.
The existence of Varg Vikernes in the early scene has had disastrous consequences and there are a lot of artists to avoid, but I still think this statement is somewhat unfair to where black metal is today.
People have learned to apply the NSBM label extremely liberally as a way of isolating those artists from the rest of the scene, but just because no one talks about RABM (red/anachist bm) doesn't mean it's not there, just that there's less urgency in pointing it out. To my knowledge (haven't looked in a bit admittedly) Mgla doesn't really have abhorrent statements or lyrics, but it's widely labelled as NSBM because of interaction with individuals who are more clearly NS-adjacent. It's healthy to do this because it's extremely helpful for listeners who want to avoid supporting that stuff - we're at the point where if there's any sketch and the group is not unknown, you'll probably find out pretty quickly. Going in the other direction though, there's no urgency to label something like Obsequiae as RABM under the same logic. Knowing that they are leftist is an interesting fact and there will be people who care, but it's very different from a dire warning that it needs to be avoided. It's just not going to be talked about with the same emphasis.
Using the NSBM label very liberally has been increasingly effective in creating a schism between the NS groups and the artists who don't want to associate. Neige (Alcest guy) I believe went to high school with the Peste Noire guy and they collaborated on stuff as teenagers including Alcest. Neige put out a statement a while back to apologize and condemn that. To my knowledge his own words had never been problematic, but more and more artists are explicitly picking sides.