Weiss explicitly talks about how her family had been targeted by the White Fang, and how that led her father to take it out on his family, in V1E15
Okay, I'm going to walk through the entire arc with you.
The episode opens with the protagonists going to the docks to meet with students visiting from Vacuo. By itself this is honestly mostly just kinda a weak padding opening so i'm not going to touch too much on it. However, they then hear talk about the heists and how it's the parties think it was the White Fang's responsibility. This leads to a debate about the White Fang. And it starts out being about the White Fang, but then it takes a turn when Weiss says this.
Weiss: "That still doesn't change the fact that the White Fang are a bunch of scum." (to Blake's growing anger) "Those Faunus only know how to lie, cheat, and steal."
Yang: "That's not necessarily true..."
Now, the voice acting in the early seasons is especially not great, but Yang's VA does a pretty decent job of sounding distinctly uncomfortable about where the conversation is going in a way that as an outsider to the conversation she didn't before. If this was still solely about the White Fang, why would she feel that way? What is the point of having
Yang say this line at all, and not Blake, thereby continuing to emphasize the conflict between the two of them? This is especially so since it isn't like Yang ever indicates any particular sympathy for the White Fang as it currently exists. that's just not part of her character, so establishing that isn't relevant. And if this dispute is about the White Fang, why the noun shift there? Why is it "Those Faunus" only know how to lie, cheat, and steal, and not "those terrorists"? Or "The White Fang"?
Both of these elements, meanwhile, make perfect sense if Weiss isn't just talking about the White Fang anymore. If you replace "White Fang" with "BLM" and "faunus" with "black people", this actually sounds perfectly like something your racist relative might slip when discussing racial issues. In the thesis being raised by this side of the argument, this is wholly by happenstance, I guess? But I'll actually give the writing staff enough credit to say that this is entirely deliberate. This is Weiss having a moment as a privileged girl who doesn't know any Faunus and engaging in racial stereotyping, something that Yang is uncomfortable with. Now things that seem a bit awkward make total sense.
We then get a bit featuring Sun where Sun seems to live up to every stereotype that Weiss ascribed to "Those Faunus" just in that very scene. Which in itself is massively tone-deaf, but that's neither here nor there. We then get some stuff with Penny that feels like it belongs in a different episode entirely but was just stuffed in here. It's not that relevant to Weiss' arc, so I'll be ignoring it.
The episode then starts to find its focus again when we go back to the Sun stuff, and this happens:
Weiss: (realization hitting) "Wait a minute." (walks over to Penny and grabs her by the shoulders) "If you're here for the tournament, does that mean you know that monkey-tailed... rapscallion?"
...
Weiss: (holding up a poor drawing of the criminal-in-question) "The filthy Faunus from the boat!"
Blake: (off-screen) "Why do you keep saying that?!"
Weiss: (turning her attention from Penny to Blake) "Huh?"
Blake: (walking over to Weiss, angry) "Stop calling him a rapscallion! Stop calling him a degenerate! He's a person!"
Weiss: "Oh, I'm sorry." (releases Penny and motions to objects around her) "Would you like me to stop referring to the trash can as a trash can? Or this lamppost as a lamppost?"
Here we have Weiss call a Faunus, somebody she doesn't know, hasn't met, has no relation to, or have any real reason to think is associated with the White Fang, "filthy" and a "rapscallion", something Blake clearly takes offense to. It's worth noting that at this point, she has no reason to think Sun has anything to do with the White Fang as far as the viewer can tell. She's just making an assumption based off, well, seemingly the fact that he's a Faunus. And Blake doesn't know who Sun is, so she has no real particular reason to be so incredibly offended on Sun's behalf if this was
just about him. And "filthy" is again, in some ways really borderline racialized language. I've sure we've all heard about "filthy foreigners" before.
We're very steadily departing from the contention between Weiss and Blake having to do with the White Fang, and a lot more with it having to do with the Faunus in general.
Blake proceeds to explicitly call Weiss out on exactly this.
Blake: "You are a judgmental little girl."
Weiss: "What in the world makes you say that?"
Blake: "The mere fact that you would sort that Faunus boy with a terrorist group solely based on his species makes you just as much of a scoundrel as you believe him to be!"
Now, by the reading you and other posters in this thread are ascribing, Blake would be wrong here. But Weiss is pretty clearly the one that is supposed to be at least primarily in the wrong in this entire episode, the way the entire episode revolves around her arc seems to pretty firmly establish that. From what I can tell from the episode, we're supposed to be on Blake's side when she calls Weiss out, not going "Oh, well, gee, I assume she has her reasons." Hell, it would make me feel even
worse about the episode if we were! The argument continues, and Blake gets even more blatant.
Blake: (
gets on her feet as well) "There's no such thing as pure evil! Why do you think they hate Humanity so much? It's because of people like
Cardin, people like
you, that force the White Fang to take such drastic measures!"
Weiss: "People like
me?"
Blake: "You're discriminatory!"
Here Blake equates Weiss with the openly racist character, and talks about how she's discriminatory, just outright. We then get Weiss' bit about how she's a victim that you quote the end of. I'll come back to it later. Ruby sympathizes, and then Blake outs herself as a former member of the White Fang. Blake then meets up with Sun, and that is basically where the episode ends.
Now let's move onto the next episode.
We open with a conversation between Blake and Sun about the White Fang.(1) It isn't too important here, really, so we'll move on. Weiss, Yang, and Ruby are looking for Blake, and then Penny reveals that she thinks blake is a faunus. Then a bunch of other stuff happens that isn' really relevant to Weiss and Blake's relationship. There's a fight scene, and so on. Then the big resolution happens and Weiss says that she doesn't care about whether or not Blake was in the White Fang but that she hopes that Blake won't keep these things secret again.
Then, however, she turns to Sun.
Weiss: (as the five gather with each other, she points accusingly at Sun) "I'm still not quite sure about how I feel about you!" (Sun laughs nervously)
this is really kind of weird. Sun has literally nothing to do with the White Fang. He hates the White Fang. He has no sympathies for them. So why is Weiss still "not sure" how she feels about him? How is this related to her revelations about Blake, as she implies with the framing of that line, given that he has nothing to do with the White Fang? The only thing that they have in common is... well... I think you know.
And then we get the end scene with Cinder and that's basically it. Notice something that I haven't touched on a single time in this synopsis? Weiss' relationship with her family or her family's relationship with the White Fang. That's because aside from that one line by Weiss, it literally never comes up. Ever. Weiss doesn't talk about how her father mistreated her as a result of the White Fang's attacks to the others, she doesn't even imply anything about it. There's no foreshadowing about it that could be called back to later. There is no connection beyond the Schnee Dust Company getting attacked by the villains, which isn't really relevant to the subplot. It's entirely incidental.
Hell, let's look at the line again.
You want to know why I despise the White Fang? Why I don't particularly trust the Faunus? (leaning against the bookshelf by the window) "It's because they've been at war with my family for years. War, as in actual bloodshed. My grandfather's company has had a target painted across its back for as long as I can remember. And ever since I was a child, I've watched family friends disappear; board members executed; an entire train car full of Dust, stolen. And every day, my father would come home, furious. And that made for a very difficult childhood.
The explanation here doesn't depend on Weiss' relationship with her father, it's a footnote in the entire thing. There's no real reason to even think from this that it was her father being mad that made the events difficult, and not the events themselves. This is exactly what I mean when I say that Weiss' abuse is backfilled. There was time to flesh it out earlier on but they deliberately don't do so.
But wait a second, what was that first sentence again?
You want to know why I despise the White Fang? Why I don't particularly trust the Faunus?
Weiss says it herself. This isn't just about the White Fang to her. This is about the Faunus. Her statements that seemed prejudice toward the Faunus were in fact, about the Faunus. Everything that indicated that she had issues with the faunus wasn't the writers accidentally doing all this, it was them deliberately doing it.
In sum, this reading of Weiss' character simply doesn't hold up. The episode is not about her getting over her hang-ups with members of the White Fang and is explicitly about her getting over her racism. The things Weiss says indicate it, the way others react indicates it, the thrust of Weiss' very own defense indicates it.
The end.
1. This is a complete aside, but the fact that this is why Sun, the first sympathetic character who we meet as a faunus, dislikes the White Fang when we're still being introduced to them and our view of them being shaped feels like it should maybe shape our perception of how rwby feels about violence means to upset the status quo.
Sun: "Of course! I don't think there's a Faunus on the
planet who hasn't heard of them. Stupid, holier-than-thou creeps that use force to get whatever they want. Bunch of freaks, if you ask me!"