grammar - What is the accepted stance on using "they" in a singular form?

The following quotes are from the wikipedia article.
It seems to me that they all use "they" for a generic person.
For example, in the Chesterfield's example: "If a person is born of a . . . gloomy temper . . . they cannot help it.",
"a person" appears to be singular but it represents any person.
It is essentially plural.

The following are similar examples.

'Tis meet that some more audience than a mother, since nature makes them partial,
should o'erhear the speech."— Shakespeare, Hamlet (1599);

"If a person is born of a . . . gloomy temper . . . they cannot help it."— Chesterfield, Letter to his son (1759);

"Now nobody does anything well that they cannot help doing"— Ruskin, The Crown of Wild Olive (1866);
"Nobody in their senses would give sixpence on the strength of a promissory note of the kind."— Bagehot, The Liberal Magazine (1910);

"I would have every body marry if they can do it properly."— Austen, Mansfield Park (1814);

Caesar: "No, Cleopatra. No man goes to battle to be killed."
Cleopatra: "But they do get killed"
—Shaw, Caesar and Cleopatra (1901);

"A person can't help their birth."— W. M. Thackeray, Vanity Fair (1848);

"When in the Course of human events, it becomes necessary for one people to dissolve the political bands which have connected them with another . . ." —United States Declaration of Independence;

On the other hand, I think the use of singular they in the following example is grammatically incorrect because it refers to a specific person hence it is essentially singular.

Someone was approaching my room.
I could see that they were alone judging from their footsteps.
They knocked on my door. I didn't answer. They knocked again. I still didn't answer so they left.