grammar - Use of singular they for specific person

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.

'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;

My question
Is the use of singular they in the following passage grammatically correct?

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.