In empathizing with someone, you don’t necessarily have to relate to their beliefs. Rather, you can relate to the universal emotions and experiences that drive those beliefs: fear, jealousy, disenfranchisement, anger, etc.
As you pointed out in your hyperlinked article about Baldur’s Gate’s villain, one of the reasons villains are written sympathetically is to humanize them. If you’re going to talk about real-life antagonists, such as your White Supremacists example, I mean, shouldn’t they absolutely be humanized because they are humans existing in our real world? Their belief systems may be awful, but these people are still products of our real world whose unfortunate beliefs are likely fueled by unfortunate circumstances, whether it’s a lousy upbringing, propaganda within their community or whatever. Again, empathizing with them does not mean agreeing with them or even entertaining their beliefs as correct; it can just mean acknowledging and relating to their experiences and emotions that fuel those beliefs.
In your Baldur’s Gate article, you describe the sympathetic villain trope as “traditional storytelling,” but my impression has always been the opposite. At least in pop culture, I always thought the sympathetic villain was a more recent creation meant to subvert the moustache-twirling caricatures of yesteryear. In reality, I’m sure different storytelling traditions from different cultures have different histories and lineages for the development of villain portrayals. However, regardless of which is more traditional than the other, I do think the sympathetic villain being so in vogue has produced a lot of contrived and overly sentimental attempts at humanizing villains.
With such an over-saturation of sympathetic villains, it totally is refreshing when you get a sensational, unambiguously evil antagonist, as you described in your Baldur’s Gate article. Sometimes you just want to really hate a bad guy. However, I don’t think one style is necessarily superior to the other. Which to use should depend on the world of the story and what the creator is trying to communicate.
When it comes to the Mindflayers, being an entire species, I’m sure I can relate to the institutional forces that have created their enslave-all-the-things, rule-all-the-places belief system. They’re not my beliefs, I’d never hold them as my own, but as a product of my own society, I understand how hard it can be to think outside established belief systems.
However, I also understand the allure of crafting an unambiguously evil Mindflayers, a sort of ancient and uncomplicated villain everyone can rally against.
I would just tell DMs to do whichever suits the story they want to tell, and not to worry that there’s something wrong about empathizing with the emotions and experiences of the bad guys.