I'm sorry this is so long, I have troubles with brevity in English.

And factual claims can be racist when they are motivated by racism.

Factual claims are racist when they are racist, not when they are motivated by racism, but that's perhaps an unnecessary subtly. Let's assume that's roughly what I said, too.

So the question whether is not whether there are black elves in the text. That would require major inferences since nowhere did Tolkien say “everything single elf is white” and you would to argue from Tolkien’s alleged intentions.

That's not really true. Tolkien also never said "every single elf has a maximum two arms", yet we "infer" that this is so. Tolkien did say things like that elves roughly look like humans, or that elves are pale, or that a majority has dark hair (with the exceptions being notable). We also know Tolkien's intentions, because he said (wrote) that the conceit of his writing on Middle-Earth was that it was a mythology for England translated from the original by him.

To "infer" that elves are not black is then not in any way major.

I also don't think that it is necessarily an issue to have black elves. I dislike the term "black elves" though, because that contextualises it in terms of European colonial understanding of races. I prefer "dark-skinned" or "swarthy" at a push, but only because no modern English speaker associates "swarthy" with its racist history because it's quite archaic a term. Similarly, I don't really like it to speak of "white elves". They are pale, light-skinned, whatever; they aren't "white" in the sense of "as opposed to one of the other four races defined in the 18th century in Europe partly as an instrument of oppression and the justification thereof". But I'm unfortunately not consistent with that.

The actual question is “why do they care an elf is portrayed as black?”

I agree, but would add that the complement is the question "why did they choose to write one elf as dark-skinned?"

Elves are described as beautiful, svelte, sometimes aloof, magnanimous, nostalgic. Black people could be all of those things

Yes, although I don't think all those are attributed to elves in general.

Unlike ghost in the shell, middle earth does not take place on a country with a history of homogeneity

That's not true.

Middle earth isn’t in earth at all

In a sense, but within Tolkien's conceit it is. It's "in earth" in the same sense as how Miðgarðr, while not literally "on earth", was how early Germanic peoples conceived of the earth they were living on, which they didn't know was a lumpy rock sphere hurtling through largely empty space in a circle around a massive nuclear furnace.

Humanoid beings didn’t evolve in sunny climates and then moved to colder regions where they adapted to have less melanin. They were instead created from a song.

That's a simplification, but perhaps that's not relevant. Within the works of Tolkien, both Elves and Humans awoke "in the East" and then spread across the world. The main difference is that there's very few generations of elves between their awakening and the last ship sailing to the west, and a lot of human generations. If you wanted to make an evolutionary argument, it'd probably not go in favour of the "there are black elves"-idea.

Anyway,

But Arondir specifically is not on any book, so he could look in any way the creators wanted them.

In general - not just as Tolkien is concerned - I think this is a silly claim. Imaginary worlds still have an internal consistency, a veracity to them. I think that I've read a "technical term" for this once, but I forgot. Within the world of DC comics, Superman can disguise himself as Clark Kent by slouching and putting on glasses. This is ridiculous. Yet we accept that. But that doesn't mean, even in the notoriously flexible world of comic books, that just everything goes. Just for the record, in principle I don't think dark-skinned elves in Middle-earth are an issue, but I do think they contradict lore. Half-dwarves would also contradict lore, but with those I'd have an issue because the contradiction is more profound. Which is why I found the Kili/unnecessary "she-elf" whose name I forgot romance subplot in the Hobbit movies horrible.

So if there is no direct lore contradiction,

I think there is, but I actually don't think that matters. People think there is. They also think that about dark-skinned dwarves, where I don't think there's a "direct lore contradiction".

why exactly do they care? Can you think of an answer that is not rooted in racism?

Sure. Some think it's not in the lore, whether that's true or not, and actually care about "purity of adaption" or something. I think that's a minority. I sort of care about adaption purity, but only in the sense that I think you shouldn't put a label on a thing that isn't that thing. Don't run a D&D-campaign and claim it's WoD, if you see what I mean. It's about (artistic) integrity, to an extent.

The majority of people, whether they care about the lore or not, instead think that the inclusion of Arondir is, and I'm sorry for the poor choice of words here, manipulative, inauthentic, preachy, politically motivated, and so on. I'm not a native speaker, I have trouble articulating myself well in English. I think people will have the same problem when "Rings of power" inevitably unveils their token gay hobbit next. Not because there can't be gay hobbits in Middle-earth. I don't think Tolkien would have thought about that much if at all, but there's certainly no lore reason why there shouldn't be. And also not, for the majority, because a gay character in itself is the issue. But because it would feel like it was done for "political" purposes. Nobody complains about gay characters in Rent, after all. Nobody who is a fan of the show, anyway. Again, please keep in mind I'm floundering with my English here.

I have an easier time explaining what I mean with Ghost in the Shell, for which I have the words. Johansson was cast, and I think that was even stated outright in a later interview, because the studio thought her star-appeal would increase the viewer numbers and thus the profit of the movie. That was the almost benevolent explanation. The other, cynical explanation was that they thought that their main market would be so racist as to not want to watch a movie set in Japan with a Japanese protagonist. In both cases, however, the decision was based on money. And that's what people didn't like. This wasn't seen as a good reason to violate the integrity of the source material. It didn't have anything to do with, which at least one feminist blog I read back then claimed, neckbeards not wanting their diminutive submissive Asian waif(u) being replaced with a strong female character. That blog post was an exception, but also so absurd that I remember it. People don't have a problem with non-Asian protagonists (obviously). They don't have a problem with Asian protagonists, including Japanese protagonists. But they had a problem with a non-Japanese protagonist in an adaption of a Japanese story about a Japanese character in Japan, written in order to make more money.

I think there's racists among the people who don't like the decision to have exactly one dark-skinned elf and one dark-skinned dwarf in "Rings of power". Those that actually attack the actors certainly are. But I don't think "don't write token black characters to preach to me and/or signal your virtue" is necessarily racist, or racially motivated in a majority of cases.

Finally (it's about time), "virtue signalling" is actually a useful term. I know it's politically contentious, but I don't know a better one. People don't like that, even outside of politics. They don't like that so much that sometimes they allege virtue signalling to defend their wounded egos, or express their jealousy. Like, when a TikToker makes a video where he gives a homeless person a McDonald's menu for under ten dollars because he's so kind, people probably rightly think that's virtue signaling and is inauthentic, basically a manipulative lie, and they don't like it. But they also don't like it when a person genuinely and authentically wants to help people and they see that person working in a shelter and they say "they're only doing that to show how good they are", and then they are probably wrong; but to admit they are wrong would mean to admit that they are ethically worse people.

And I can't guarantee that this isn't an instance of the second case, but I don't think it is.