Prefacing this discussion with the caveat that we don’t know for certain what is true about this story, since Arya and the waif had already established the precedent of the lying game and since the waif herself said both that there was “an untruth” in it and that she “lied about the lie”. For these purposes, however, I’m going to assume that it is more or less accurate (including Arya’s deduction that the House of Black and White took two-thirds of the family wealth, not all of it).

You’re certainly right that GRRM loves his fairytale tropes: it’s barely scratching the surface to say that he has drawn inspiration from stories like Beauty and the Beast (in the stories of both Jaime and Brienne and Sandor and Sansa), Snow White (in the story of Cersei and her fear of the younger, more beautiful queen), and Little Red Riding Hood (whose story beats get twisted in Sansa and Sandor’s meeting during the Battle of the Blackwater), among many others, for ASOIAF. Indeed, he uses Cinderella’s basic plot points for Falia Flowers as well - the daughter of the household treated as a serving-maid by her father’s wife and the woman’s daughters, only to triumph over them when she “marries” a king - but, as he does with many other tropes, plays with and subverts them, having Euron reveal his truly evil nature in his treatment of Falia afterward.

The waif’s story, in that sense, also borrows elements from the classic (at least, to a western audience, appropriate given GRRM’s own background as a member of this audience) telling of Cinderella. The waif’s father’s second wife fits the Wicked Stepmother trope all too well, and if the woman’s daughter isn’t strictly a wicked stepsister (she seems to be the child of the waif’s father as much as the waif herself is, and we get nothing of her character in the story), there is certainly a sense of prioritizing the daughter of this second marriage over the daughter of the first. Perhaps GRRM was also thinking of the Grimms’ more violent (second) telling of the tale (certainly compared to the Disney film), in which the wicked stepsisters are initially mutilated (on their mother’s advice, in order to fit their feet into the all-important slipper) and later blinded as punishment for their treatment of Cinderella; certainly, the woman’s literally poisonous attempted murder of the waif, and the father’s willingness to have the woman murdered in turn reflect a distinctly non-sanitized take on fairy tale tropes. Again, because nothing is one to one between ASOIAF and its sources of inspiration, not all of the story beats of Cinderella are found here: the waif does not live happily ever after as the bride of a prince but is almost poisoned to death before being forced to spend the rest of her life working for a cult of assassins, and she seems to have none of the sweet, kind, unfailingly good nature of the fairy tale heroine.

In universe, the story underlines just how different the waif is from Arya; indeed, they are almost polar opposites in their family upbringings. The waif was “born the only child of an ancient House, my noble father’s heir”, her sole sibling eventually being a stepsister and rival for her dynastic place; Arya may also have been born to “an ancient House”, but she grew up in a happy, relatively sizable family of five siblings, mixing the children of Ned and Catelyn with the (ostensible) bastard son Jon, and as the younger daughter had no worries about being prepped to inherit Winterfell. The waif speaks of never knowing her mother, who died when she was young; Arya, by contrast deeply loved and misses her mother, hates that she could not save Catelyn from the Red Wedding, and even seemingly named herself after her mother when prompted for a false identity in Braavos. The waif was subjected to the murderous ambition of her stepmother, who sought to drive the waif out of the family inheritance in favor of her own daughter, yet I tend to believe that the Stark kids, happy to be reunited with their loved ones, will resist factional attempts to pit them against each other for control of Winterfell (with the obvious reminder that there is, you know, an apocalypse coming). The father of the family has no problem seeking murderous revenge in turn against his second wife, though this could not be farther from the truth about the relationship between Ned and Carelyn; indeed, Arya so much believes in the love between her parents that she angrily defended it to Edric Dayne and included the same sort of parental love in her invented backstory as Cat of the Canals. Nor could we ever believe that Ned would sacrifice his daughter to a cult of assassins for the sake of revenge; rather, we saw Ned’s willingness to renounce his firm beliefs about the succession and condemn himself to life imprisonment (and, though he might not have known it at the moment, death for himself) so that his daughter might be kept safe.

All of these contrasts emphasize that the House of Black and White is not the place of Arya of House Stark. This is a place which not only profits from dissension, infighting, and murderous ambition and revenge, but indeed depends on these to form its “market”, so to speak - a far cry from the happy home of Winterfell and its Stark inhabitants. Where the waif treats the implosion of her family life as merely a new entry in the lying game, Arya has never stopped remembering, or missing, her life among her Stark family members. Arya cannot give it all up, as the waif did (or, rather, as the waif had it decided for her that she would); she belongs back with her family and back in Winterfell.

Of course, the waif’s story is full of ironies as well. In the end, the grand fortune of that “ancient House” the second wife might have hoped for her daughter to inherit all but disappears into the coffers of the Faceless Men; only a third is left, less even than what the second marriage’s daughter would have gotten had the inheritance been divided equally. The father is left back where he started, dynastically speaking - the single parent of a single daughter - yet poorer for it, arguably no better and perhaps worse off than he had been before (only limited by our lack of insight into his personal view on the matter). If the father’s goal was to murder his second wife in revenge for what she had done to his elder daughter, his actions simultaneously took that daughter’s life as well; the waif was forced to spend the rest of her days forsaking her entire birth identity and murdering others in the name of an assassination guild, legally and spiritually if not bodily dead. (Nor, of course, do we ever get any insight as to what the waif herself, the actual target of that attempted murder, thought of either the contracted murder of her stepmother or her own permanent confinement in the House of Black and White; in an act of revenge ostensibly taken in her name, the waif has no input and no agency.) Likewise, while the father had his second wife murdered, he seems to have done nothing to disinherit the child of his second marriage. This daughter was in turn left precisely where the waif was, her father’s only heir and perhaps too young to remember her dead mother; the stepmother’s ambitions posthumously won out, as her husband’s elder daughter was effectively killed off and her own daughter was set to inherit. All of these ironies add to the waif’s characterization, making her an outlet of narratively rich writing as opposed to simply an NPC Arya interacts with in her new environment.

On a practical level, too, the story provides more insight into elements of the Faceless Men’s business model. The Faceless Men aren’t out to completely beggar their clients - hence asking for two-thirds of the man’s wealth instead of all of it - but they do ask for a considerable sacrifice, at least within the means of the requestor. The father in this story was probably a wealthy man (the head of that “ancient House” in a society which has historically valued trade and commercialism over feudal land ownership and agriculture), so the price was high in monetary terms - but because the father was also a man in the midst of a succession dispute, the price included the previous guarantor of his dynasty, his elder daughter (with, again, the implication that it would be the child of his targeted wife who would succeed him). Perhaps this pricing model was intended to imbue this killing with the divine approval of the Many-Faced God: if this man were willing to have his wife killed knowing that by his actions he would disinherit his older daughter and impoverish himself to some extent, then truly this request must be the will of the Many-Faced God, and not simply human vengeance.

Plenty Ned could do to avoid his execution as in canon, and it goes further back. Ned was not locked into failure. Cleaning house in the Small Council well before that point was something he could and should have done. Or appointed more people to balance out the toadies. Shifting the balance of power in King’s Landing was well within Ned’s ability as Hand.

Even afterwards, forget not warning Cersei, if Ned had followed up on warning Cersei. Hell, if he’d followed up more aggressively with Stannis - Ned’s Hand of the King, he can send letters like “show up at court or provide a good explanation why not” - Ned could have been weeks ahead of himself.

Sansa’s actions don’t have much to do with Ned’s capture, I don’t think - they result in her own, yes, because neither she nor Arya could get on the ship planned to take them from King’s Landing, but Cersei was already planning to move against Ned when Sansa provided the info, and Ned told still more to Littlefinger afterwards.

Politically, Ned’s fundamental mistake is that he doesn’t realise that he is the boss. Between Robert’s inattention and Robert’s regard for him, Ned can do pretty much whatever he likes as long as it doesn’t bother Robert himself. Do we really think Robert’s going to go to Ned “whoa whoa whoa, you can’t just sack the treasurer!” and make that stick?

But even with everything, it’s simply Ned’s bad luck that he spends a bunch of critical chapters drugged as hell to manage the pain from his broken leg. The author kneecapped him so he couldn’t get away.

He went on to tell how the remnants of Ser Stafford’s host had fallen back on Lannisport. Without siege engines there was no way to storm Casterly Rock, so the Young Wolf was paying the Lannisters back in kind for the devastation they’d inflicted on the riverlands. Lords Karstark and Glover were raiding along the coast, Lady Mormont had captured thousands of cattle and was driving them back toward Riverrun, while the Greatjon had seized the gold mines at Castamere, Nunn’s Deep, and the Pendric Hills. 

Catelyn V, ACoK

Now, even in this passage, there’s differences. The raids Robb has been conducting are aimed at Tywin’s supply - coastal towns, cattle, and gold. There’s targeting there, some evaluation of their military value. As opposed to Tywin’s policy of “see a field, burn a field” policy in the Riverlands, which is purely inflicting suffering on civilians because fuck the person who’s nominally in charge of them. Robb’s raids cause collateral damage and that’s bad; with Tywin the pain of ordinary people is the point and that’s worse.

Also, to be blunt, Tywin started it. It sounds petty, but it’s another important note with Tywin. His actions force escalation and damage the chances of restoring peace. Given the chance, Robb would be leaving well enough alone. Tywin, not so much.

What I think the narrative is ultimately trying to say is that Robb is a different person to Tywin - but what I see here is that it doesn’t make much difference to random smallfolk whose lives have been upended. Or just ended.

Tywin and Robb can be different, with Robb clearly a better option than Tywin, and they can still both be doing things that are bad for the common people.