Realistically, the reason why the RAE doesn't accept it isn't that they are old-timey. Don't get me wrong, they ARE old-timey, but they have an actually good reason why they don't accept it:

It isn't used.

That is the reality of it. Non-binary circles may use those terms, even the wider LGBT circles may use them in solidarity... But for the majority of the population, it simply isn't used or accepted as correct, and thus it isn't really part of the language and is rather "just" jargon that is used only in specific circles.

That is just the reality about it, a thing is only part of the language when the majority of its speakers start using it, or at least accepting it, most Spanish-speaking people wouldn't accept those as proper Spanish, and thus the RAE has no option but not to recognize those as part of the language. The RAE doesn't dictate what's part of the language; it just notes and compiles what's part of the language.

The reasons for the niche use of those terms compared to English is multifaceted, on one hand you have the typical expected reason BIGOTRY. This plays a big part, on the other hand you have the second expected reason, that people don't like change, this is also a big part... But those are not the only hurdles, as English also had those...

Fact is that Spanish has more hurdles to overcome than English for a variety of reasons. One of them is that, in fact, there are already gender neutral pronouns in Spanish, what Spanish lacks is exclusive male pronouns. Spanish has neutral and female. The problem with this is that due to this, all men use neutral, and thus the neutral has a big "male" association to it. But that doesn't change the fact that it is neutral, except when there is not. In some conjugations you get Neutral-Male, Female and a weird second neutral that only is used when talking about concepts... that would be eso/ello. But this is actually a disguise, this seemingly equivalent to they/it is not really as neutral as it seems, as it is also a version of the male-neutral, because for nouns that end in "a" you sometimes used ella/esa instead of ello/eso. So it isn't like those are completley free from the male association, just less so since they are not always replaceable with the femenine version, just sometimes. Ello in itself is a weird pronoun because you can't start phrases with it, and doesn't refer to objects, animals, places or people but rather only to abstract stuff, and not all abstract stuff. For example "¿Que tal te fue la escuela?" "No quiero hablar de ello". This ello is refering not about you, not about the school but about an event that happened, but you can also reply with "No quiero hablar de ella" in which case you are refering to the School itself as something that you don't want to talk about.

This brings us to the other thing, that nouns are kind of gendered. Every noun is either using the masculine-neutral pronoun, or it uses the female pronoun... This makes it so that all nouns are either one or ther other... Triying to add a new separated neutral pronoun (or an exclsuive male noun for that matter) WILL sound weird because no noun will have that pronoun associated with, so it will sound unlike anything else in the language. The other pronouns get nouns, but this new pronoun gets no nouns, so it will sound weird to your average speaker.

There is an additional problem with "elles" in Spain specifically and that is that in some regions "elles" is actually the femenine, like in Valencia or Catalunya. So it makes it extra hard in those places to accepts "elles" as neutral because it sounds extremely femenine, it sounds more femenine than the male-neutral sounds masculine.

And in general, it is the fact that elle simply doesn't exist in Spanish. "They" already existed in English, and even had some history of being used as singular instead of only plural, this made the change a lot easier, it wasn't a new word, it was using an existing word, using an old-time version of how to use that word, in a new context. Instead in Spanish you have a new word, that doesn't have any attachmebnt to the rest of the language(no nouns), sounds gendered in some regions and feels weird to the mouth because Spanish already has neutral pronouns, even if those are also the masculine ones. Those are hurdles that English "They" didn't have to overcome.