Neutral Spanish isn’t a separate variant, so much as a separate register of the language, though. It’s really just a thing I hear native speakers say when they don’t realize that educated speakers from their country do, in fact, still have an accent, but it’s more just down to vocabulary choice, rather than some major change elsewhere. Like, an educated Dominican isn’t going to call a bus a guagua and they’ll probably enunciate more clearly than they would in casual conversation, but they’re not suddenly going to start using vosotros and distinción when they speak.
Whenever I hear a native speaker talking about Neutral Spanish, it’s invariably followed by why I should try to speak like people from their home country, and that people from elsewhere don’t really speak proper Spanish. It also tends to correlate pretty well with people telling me, “Yo hablo castellano, y por eso no puedo entender lo que dicen las personas plebes, ya que hablan español.” for a nice dose of Latin American classism.
If you learn something too region specific, usually doesn’t.
My experience has been more that learners tend to not realize that certain things they pick up aren’t universal, and/or that they’re only acceptable in certain contexts, and then unwittingly pepper their speech with words and phrases from one country that are unknown/unacceptable in another, or use very informal/vulgar language in formal settings. Like, if I curse around my wife the way I would curse around my Mexican coworkers, she’s scandalized by how vulgar the profanity is, and if I told my Mexican coworkers I had a fuinfuán in my backyard growing up, rather than a columpio, there’s nearly 100% chance they’re not going to have any idea what the hell I’m talking about.
I’m not really concerned about it myself, I’m already well beyond the point it would be terribly relevant to me. I would very much disagree re: education levels, though. I’ve worked with plenty of people from various countries, and those with less education often cannot switch to a more widely understood way of speaking, in my experience. Partly, it comes down to limited vocabulary, resulting in them being unable to provide alternative ways of saying things that might be more widely understood, and partly down to an ignorance as to what elements of their speech aren’t widely used or understood outside of where they grew up.
I would still argue that a neutral Spanish is no more real a variant than BBC English or “General American” accents and mannerisms used by news presenters represent actual variants of English, though. It might serve as a crutch for intelligibility in cases of extremely heavy accents, but most cases where you might employ it are situations where you already wouldn’t be expected to employ much in the way of slang. In regular interactions, though, people mostly just speak to each other in their natural accent, and if somebody busts out a local term that isn’t understood, the other person asks “¿Qué quiere decir huachicolero?” gets an explanation, and the conversation moves on, same as it does in English.
At the end of the day, I think pursuing a neutral manner of speaking from the beginning is something of a fool’s errand for most language learners. Like it or not, you will wind up speaking like the native speakers you interact with most. I don’t particularly use Dominican vocabulary, but people still assume I’m from DR when I speak Spanish, because when Spanish became my primary working language for 5 years after getting out of the beginner stages, that’s who I was surrounded by at work. Absent very specific goals (I knew a guy who focused exclusively on Rioplatense Spanish, as he was moving to Argentina in a couple of years to study in Buenos Aires), I think most people would be better served focusing on the fundamentals, reading widely, consuming a wide range of media and actually speaking with people, rather than endlessly agonizing over perfecting the process before actually getting to the point they can actually use the language.
After years of regular use, I can speak it fine and modify how I speak appropriately, as the situation calls for. If it’s sufficient for the RAE folks working on the DELE and the staff at my local Instituto Cervantes to not remark on anything beyond occasionally flubbing the gender of a word, I’m not too worried about the neutrality of my Spanish.