Is “they” a correct gender-neutral pronoun?

Yes, it is absolutely perfect. Good luck.

It's just as "correct" or "incorrect" as any other piece of modern grammar. The thing to remember is that grammar (and grammar rules) change over time; we don't use middle or old English grammar rules anymore, do we?

So my short answer is that yes, it's fine to use "they" as a gender-neutral pronoun.

You may also have heard of the two alternatives:

"he or she" - while this is used, it's noted as being rather awkward. Most SAT courses will still tell you to use this, but it's not strictly loved in most circles.

"he" - for a while, the masculine pronoun was used for a gender-neutral (or more specifically, gender-unknown) pronoun. This practice is not limited to English: in Latin, the pronoun used for an unknown gender (or a crowd of mixed genders) is always masculine. A crowd of students in Latin is "discipuli" (the masculine), not "discipuli et discipulae". This holds true even if there is one male student in a class of 50 females. The use of the masculine pronoun in this way is now considered politically incorrect.

Well, it's certainly not perfect, but it's become so prevalent there's no going back. Some of us older than dirt still gag when we see 'they' used as a singular. Part of that is probably because its use came from a politically correct agenda that had nothing to do with language or accuracy.