Learn English – “Does” with singular they

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.

Although there are many questions here which deal with grammatical number and plural subjects, this is the first time I've heard of this "rule". It deserves its own answer.

You are entirely correct. A plural subject requires a plural verb, especially when each "element" of that subject might be considered an individual.

The only time where an apparently plural subject takes a singular verb is when that subject is considered a single entity in its own right:

Fish and chips is the best meal ever devised.

It's possible that whoever told you that erroneous rule has come across a sentence like that and simply come up with a reason which fits it (fish and chips may both be extracted and the sentence becomes singular).

But it makes no sense: if you remove Martin or John from your sentence, you are only talking about the remaining person. Any verb in that sentence must be singular. To extrapolate that and say that because each element is singular, both together can be is frankly ridiculous. Your intuition that it is ridiculous is correct. If a subject consists of more than identifiable and separable element which are not taken together as a composite entity, it must be plural.