One misconception: They/them has not been strictly plural for quite a long time. Even Shakespeare used it.
There's not a man I meet but doth salute me / as if I were their well-acquainted friend
(From Shakespeare's The Comedy of Errors, written sometime around the late sixteenth century)
Using they as a singular pronoun is perfectly fine, especially when you don't know the gender of the person (or when they are non-binary).
Historically, of course, "they" is plural. That is why it goes with the plural conjunction of verbs - "they are" and not "they is". However, the same can be said for "you" (which was also historically plural, with the singular being "thou"), and I don't think any modern speaker would ever argue that "you" cannot be singular. This just to say, history does not dictate modern language. Many authoritative English guides accept it (at least informally), so if you look to them, then you should feel free to use it.
On another note, one thing that doesn't yet have a consensus is whether the singular referential should be "themselves" or "themself". The former is the historical one, the latter the more logical one.