It’s not only that we lack a gender-neutral pronoun. We lack a pronoun with a suitable amount of indefiniteness to properly refer to antecedents such as “anyone” and “someone”. It’s not just the gender that isn’t specified; it’s all markers of specificness. On that score, “they” is little more than a workaround. But at least we can point to other examples where “they” also has an extended secondary sense of the indefinite: in “That’s what they say” and “Give ’em hell”.
Surely everyone would admit that “singular they” has been used in enough reputable writing to be grudgingly referred to as a legitimate variant–and not an outright error. Furthermore, if you outlaw “singular they”, only outlaws will use “singular they”.
Don’t you agree that it is not a question of singularness but rather indefiniteness? Quantified pronouns like “anyone” and “someone” are notionally more plural than singular, but in English we only have two choices for subject-noun agreement for number: singular or plural. One had to be chosen, and it was singular. What we need is a new system to accurately handle the indefinite number.
Furthermore, tens of thousands of English nouns have extended secondary meanings and functions. What’s the big deal about “they” being extended to handle an indefinite antecedent? So far it’s the best we have. The thought of a manufactured replacement sickens me–but I’m not worried since “they” is extremely well-established in both usage and prescriptive scholarly acceptance.
“Themself” is attested in edited prose, but has fallen out of use. Such a bold juxtaposition of traditionally plural (though obviously indefinite) “them” with singular “self” challenges even my libertine grammatical sensibilities. Of course you will now see “themselves” used in its place. If you could accept “them” as the referential equivalent of the indefinite quantifier “any” then “anyone” and “themself” seem a match made in heaven.
“Everyone” has a distinctly non-singular flavour. And though they have “one” in them, the indefinite quantifiers prevent these pronouns from referring to a single specific entity. We treat them as singular out of syntactic convenience only.