I mean honestly, the morals of this feel pretty cut and dry to me. "Does this game actually promote the views that Rowling is criticized for?" As far as I've heard, the answer (at least for those views relevant to this discussion) is no, and Rowling wasn't even actively involved in its production, she just gets a cut of the profits because it exists in her IP. Its only [relevant] 'sin' is that it is set in her franchise. That's it. As such, criticizing the game for those views is misaimed, amounting to little more than guilt by association.
...Or really, guilt by association with a group whose founder has endorsed prejudicial views (views that the group does not endorse) outside of that group. ...You know, the exact same logic we rightly mock when Pro-Lifers try to poison the well by claiming that Planned Parenthood's founder was racist? Or when the usual suspects try to make claims about BLM's founders that they insist necessarily invalidate the entire movement? Or when creationists make allegations about Darwin as a person as if that would undermine evolutionary theory? We recognize that even if those criticisms were factually correct, they'd still be irrelevant because the transitive property does not work that way. Legacies often take on an independent existence that renders their creators all but incidental.
Never mind extending such accusations to those who play the game, which is no more solid than claiming anyone who watched Mickey's Christmas Carol is tacitly condoning antisemitism on the grounds that Walt Disney himself was antisemitic (veracity of that claim about Walt notwithstanding, we will assume it to be true here here for the sake of argument). The logic simply doesn't work and is only maintained through threading a line through several intermediaries with the functional equivalent of strings and thumbtacks while pretending for all practical purposes that those intermediaries don't exist.
I mean, let's be honest here, we have much better case for connecting Lovecraft's works to his racism and prejudices because the more you learn about the guy the more you understand how those prejudices bled into those works. Hell, when you learn about his disgust towards miscegenation, it becomes all too easy to infer that Shadow Over Innsmouth was practically a "Great Replacement" allegory. Even so, we draw a clear line between Lovecraft's prejudices and his writings, instead judging the latter on an individual basis rather than dubbing them irrevocably tainted simply because of their authorship.
And even when the stories are recognizably inspired by his prejudices, we still are able to appreciate them as stories despite that inspiration. Reading and even enjoying Innsmouth is not considered a tacit agreement with Lovecraft's flaws, even those that inspired the story. And this is a guy who literally waxed poetic (in a poem I shall not name) about how he saw black people as semi-humans filled with vice whose god-given purpose was to simply bridge the otherwise insurmountable gap between man and beast! The guy was a piece of work, especially in his early years.
But despite his appreciable failings, we certainly don't judge people for playing Call of Cthulhu as tacitly standing with that contemptible poem. The TTRPG is recognized as distinct from the poem. While Lovecraft's rather pronounced personality flaws are thoroughly worth criticizing, not even the stories he himself wrote are deemed racist purely by way of the transitive property. Any argument deeming one of those works racist is instead predicated on any racist elements contained within it, and even that is rarely - in itself - treated as making the work not worth reading.
Of course, it's even more of a stretch to try - as if it were a matter of course - to apply his flaws to the works of any writers who add to the mythos he started, much less to the readers who consume those stories. It's nothing short of absurd to claim that anyone who reads August Delerth's "The Return of Hastur" is tacitly condoning the views Lovecraft expressed in the aforementioned poem simply because "The Return of Hastur" is part of the Cthulhu Mythos, which was created Lovecraft, who wrote that poem as an expression of his racism. That simply does not follow.
But that is exactly the logic we seem to be insisting on here with regards to Hogwarts Legacy and Rowling: that because Hogwarts Legacy (analogous to the TTRPG, which obviously was made without Lovecraft's involvement) exists in the Harry Potter franchise (Cthulhu mythos) - which is owned by Rowling (Lovecraft) - playing the game is treated as condoning Rowling's personal political views.
For ease of example, let's focus on your example of the streamer playing it. We're taking an insanely reductionist view that anyone who consumes the game in any form should be defined by a component that - to the best of my knowledge - the streamer is not expressing or condoning (and in fact may be demonstrating opposition to), and is neither present in the game, nor the franchise it is a part of. That we need to go down the grapevine until we get to the original creator whom we reduce to that component we are objecting to because she expressed those views as personal political speech, and then treat that as an intrinsic and irrevocable component of everything that is so much as involved with materials connected to the things she created. It's specious logic at best that basically tries to pretend that a chain of indirect links through association are instead a single direct link (sometimes even as far as a causative one).
Even if we're trying to argue this through Rowling's personal finances going to organizations that share/champion those views and us adding to those finances by consuming (in any manner) anything she gets royalties on, that still hits the problem of trying to present a chain of indirect links as a single direct and clear link. That if you in any way touch Hogwarts Legacy, you are making it clear that you support those prejudicial agendas by supporting those groups purely because Rowling gets some money through the game by way of royalties and some of her money goes to those organizations.
One might as well be arguing that anyone who watches Top Gun Maverick should stop lying and just admit that they're really a Scientologist because Tom Cruise starred in it and he's a Scientologist, and that therefore anything short of you boycotting anything he gets money from means that you are indirectly financing that organization. It's the same logic.
And again, it's specious logic fully embraces the consumer-blaming variant of the "no such thing as ethical consumption" sentiment and exacerbates it with "six degrees of separation" logic, which means we can theoretically make a similar connection to practically anything by employing a similar number of links through different people and groups.
So my conclusion is very simple: If we have to go that far down the rabbit hole to justify the objection, the link is not substantive enough to be a tenable position and certainly isn't worth policing our companions over, much less burning bridges over. Treating this game as an extension of Rowling's views despite the views not only not being represented in it but Rowling not being involved in its production isn't taking a stand against Rowling's transphobia, nor is treating the act of playing the game like declaring allegiance to those views. It's just applying total war tactics to blood feud logic and declaring that poor Romeo having the surname Montague is reason enough to not only want him dead, but to declare anyone who so much as sells him groceries persona non grata.
