As found in Wikipedia, Singular They can be used for:
Indeterminate gender – when they refers to an individual person of unknown or unspecified sex, as in, for example, "One student failed their exam". This usage is known as epicene they.
Indeterminate number – when they has no definite antecedent, or can be interpreted as referring to either a singular or plural entity. This usage is also known as generic they. For example, in "Anyone who thinks they have been affected should contact their doctor", they and their are within the scope of the universal, distributive quantifier anyone,[1] and can be interpreted as referring to an unspecified individual or to people in general (notwithstanding the fact that "anyone" is strictly grammatically singular).
It further notes:
In some cases, they is used even when both the number and gender of the subject are known, but the identity of the person is generic, e.g. "If some guy beat me up, I'd leave them."
Though semantically singular or ambiguous, singular they remains morphologically and syntactically plural (e.g. it still takes plural forms of verbs).
To specifically answer the question posed:
Your reader is [...], but they are a busy person.
is correct. Even though your reader is singular, and when replaced by the singular they pronoun, they remains morphologically and syntactically plural and thus they are is appropriate.
You, of course, could bypass the whole issue, and say:
Your readers are [...]. but they are busy people.
When common nouns are used in Subjects, the verb usually agrees with the noun. So if the noun is singular, we see 3rd person singular agreement. If it is plural we see plural agreement:
- The parrot is cute.
- Parrots are cute.
However, when a pronoun is used as a Subject, the verb always agrees with the pronoun. It doesn't matter what the meaning of the pronoun is!!
So when we use the pronoun one, it doesn't matter if it means "we" or "you" or "people", the verb is always 3rd person singular.
- One is in a bad mood today. (means You are in a bad mood today)
When the queen uses the pronoun we but she means "I", she still uses plural agreement:
- We are not amused. (means I am not amused)
In the same way, when we use they as Subject, we always see plural verb agreement. It doesn't matter if we mean "he" or "she" or "that person" or "those people". The verb agrees with the pronoun, not what the pronoun means:
- The new student is very happy. They always bring me chocolates after the lesson. (not brings!)
- The new students are very happy. They always bring me chocolates after the lesson.
The Original Poster's question
We need to use do with they. We can't use does:
- That student is unhappy. *They doesn't like maths. (ungrammatical)
- That student is unhappy. They don't like maths.
Best Answer
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
It is actually explained in the article you linked to:
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":
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.