I've read this article in The Atlantic saying that in English schools in Europe, teachers have started teaching pupils the new "they" used as a gender-neutral or genderless singular he/she/it.
Is it true? Does this come from having any strings leading into deep history of English where this existed?
To start, the sentences you gave are not how singular they is normally used. The verb simply takes the normal form for they. The sentences would be:
As for
Does this come from having any strings leading into deep history of English where this existed?
It is actually explained in the article you linked to:
[W]e tend to miss that English speakers have been using they in the singular since English was anything we'd recognize as English. Back in Middle English, the Sir Amadace tale includes, “Each man in their degree.” The Bard has Antipholus of Syracuse in Comedy of Errors chirp, “There's not a man I meet but doth salute me / As I were their well-acquainted friend.” Thackeray has Rosalind toss off in Vanity Fair, “A person can't help their birth.” Whence the idea that all of these people were butchering the language?
So, yes, this use of they is quite old in English. The article goes on to explain why people started to think it was "wrong":
It was the schoolteacher and writer Anne Fisher whose English primer of 1745 began the notion that it's somehow bad to use they in the plural and that he stands for both men and women. Grammarians of Fisher's day tended to believe that real languages should pattern themselves after Latin and Ancient Greek, in which the words for they happened not to have experienced such developments.
Like so many nonsense-rules in English, that sadly have been taught to generations of students, the whole notion came from the misconception that English should be some form of Latin or Greek. "Never end a sentence with a preposition" is another example of those grammatical fancies that were drilled into the heads of unsuspecting students without any good reason.
I think the title Call Them What They Wants was chosen for a joke, to emphasize the fact that the singular they is not used everywhere and is even rejected in some style manuals in favour of 'he or she'. The verb used with the singular they actually doesn't take (e)s in the 3rd person Present, and remains grammatically adjusted to plural (same as with 'trousers', as suggested earlier in the related forum.
It should be
"They get what they want"
and
"When they need help, they get it."
They has a singular case, but that shouldn't change the rest of the sentence. The key here is to recognize that you don't know who or what they is. They could be congress, or Steve from accounting. "They get what they wants" is just poor English. No amount of teaching it that way will ever fix change that, and if you leave the "s" on the word "wants" when using "they" in the singular case, you're going to sound awkward and grammatically incorrect at the most basic level to native speakers.
Firstly, as per other answers, the sentences still need the correct verb forms:
They can write what they want.
When they need help they get it.
There's nothing particularly new in using 'they' as a singular. What it denotes is that the person in question is an unknown or unspecified person. It is therefore often used in conjunction with words like 'someone' or 'whoever':
Someone who writes a diary can write what they want.
Whoever it was that wrote that rude message on the wall, I think they need help.
The word "they" could refer to "him" or "her". You could refer to "what he or she writes", "what he or she wants", or wants", or "the help that he and/or she needs".
Eventually some people tried to shorten "he and/or she" to "they".
Note: In JoeTaxpayer's answer, the same phrase could have been, "Whoever left his jacket in the library..." (instead of "their jacket").
Even for people who don't like "he and/or she" being replaced with "they", enough people have talked (and written) this way that most people will understand it easily. JoeTaxpayer's sample is a great example of how something can be wrong (really, "his jacket" would have been a better way of saying it), but sounds tolerable enough that many people may not even notice the issue.
Some people liked that. Some people didn't. The shortened version was liked by some people, and disliked by others. I believe the fair consensus is that this has is considered to be controversial. In other words, there is a lot of disagreement.
So, if anybody tells you that this is definitely "good English" that is widely accepted by most speakers, that is wrong. However, if anybody tells you that this is definitely "bad English" that is widely rejected by most speakers, that is also wrong. This "rule" of English is currently in a state of being questionable.
If you are thinking of using such a phrase yourself, the safest approach would be to think of another way of phrasing things, thereby avoiding the argument altogether.
English is fluid, and often, 'proper' follows usage. It make take time, but there's no ignoring this.
I recall, in the late 70's, an announcement came on the PA system in my high school. "Whoever left their jacket in the library, please stop by to pick it up." It was an all boys school, no need for the polite gender blurring 'they.'
I work in a HS now, and so far, it's less a matter of being 'taught' as 'accepted'. People are choosing their pronouns as a statement of their own self-identity. And in an age of acceptance, this verbal choice process is taken as a sign of not being against those who make such choices.
