It's not correct.
Well, mebbe. But if you are thinking about English as a correct thing that is separate from the people who speak and write it, you are misguided. Classical Latin is the kind of language you're thinking about. It is always correct, because it is dead and cannot change. The people who speak it are long dead, and it has morphed into Italian, French, Spanish, Portuguese, and Romanian. But English is a living language, and whatever its speakers and writers collectively make it over time is what is correct. That is why it is no longer Anglo-Saxon.
No, you're wrong; it's a rule.
"Any fool can make a rule, and every fool will mind it," Henry David Thoreau said. It became a rule in the eighteenth and nineteenth centuries when grammarians tried to "clean up" English, usually by trying to make it more like Latin. (That's when we got the split-infinitive and stranded-preposition bogus rules.) But establishing rules from the top down doesn't usually work. Generations of English teachers have done battle with ain't and might could without ever doing any better than fighting to a draw. Noah Webster got the u out of colour and honour in American English, but most of his efforts to reform spelling were failures.
I was taught that it was wrong
Have you seen the test on which a fourth-grader at a Christian school got an A for saying that the Earth is not billions of years old and people once lived with dinosaurs? Children are taught a great deal of rubbish. In your case, I am afraid that the statute of limitations has long since run out and your chances of winning a judgment against your sixth-grade English teacher are negligible.
A personal pronoun can't be both singular and plural.
So you have swallowed the camel of you as either singular or plural but you strain at the gnat of singular they?
This is just pandering to the politically correct.
There's no dispute that people who favor gender-neutral language approve of singular they. But the usage was common and acceptable in English for five and a half centuries before political correctitude. Look it up.
My dissertation adviser/editor/boss won't accept singular they.
Render unto Caesar that which is Caesar's.
It's easier to write around it than irritate the sticklers.
Do you really want to let the crazies win? Weren't you the one complaining about pandering just a couple of paragraphs back?
"It's ugly and pig-ignorant."
Nigel Cameron offered this trenchant counter-argument in a tweet about this post. What I take him to mean is that he, like you, doesn't like it. You don't have to like it. No one is making you use it. But unless some shadowy group of electors has met in conclave to proclaim your imperial authority, you don't get to dictate your personal aesthetic preferences to other speakers and writers.