Hasty generalizations about grammar quickly get you into trouble. As a case in point, consider the difference between subject-verb agreement and pronoun-antecedent agreement. Both require, in the core cases, attention to the number (singular or plural) of a particular noun phrase. At the same time, there are important differences, and treating the two as identical can lead to significant problems.

Subject-verb agreement is based on a syntactically fixed relationship between two specific slots in a sentence. Present-tense verbs must agree in number with their subjects, which are prototypically noun phrases. The number of the subject depends on the number of the head noun in the phrase. Thus, when a subject contains embedded elements like prepositional phrases, those elements are typically irrelevant in determining number. So, for example, the subject and corresponding verb in the following sentence are singular:

The consensus of all the faculty members was to accept the applicant to the PhD program.

Consensus is the head noun of the subject ("The consensus of all the faculty members"), and as it is singular, so is the complete subject. Faculty members is a noun phrase embedded within the larger noun phrase as part of a prepositional phrase and plays no role in agreement with the verb.

In contrast, preposition-antecedent agreement does not have a syntactically fixed organization the way that subject-verb agreement does. While there are some constraints on what can serve as an antecedent for a pronoun, those constraints do not prevent the antecedent from appearing in any syntactic position, including as the object of a preposition. For example,

The praise that was given to Peter only made him more egotistical.

Notice that Peter is in a prepositional phrase that is part of the subject, but it's unquestionably grammatical for him to refer to Peter.

There is nothing at all controversial about the observations above. It probably wouldn't even be a significant issue if we didn't have high school students preparing for the SAT and similar tests. These tests feature problems with putative errors in both subject-verb and pronoun-antecedent agreement. And when students are taught how to find subject-verb agreement errors, it's natural for them to over-generalize and apply the same principles to pronoun agreement. I find it less easy to forgive the same confusion in published material.

The following problem in appears Kaplan's 12 Practice Tests for the SAT (2008), p. 117. If you're unfamiliar with the question format, the task is to find which underlined segment, if any, contains an error:

28. Concerned about the playoff game on Saturday, each of the team members spent most of the week practicing their plays. No error

The intended answer is that their constitutes an error. Here is the explanation for this problem (p. 165):

The pronoun their, which is plural, is referring to the pronoun each, which is singular. If you were correcting this sentence, you would have to replace their with his or her. Notice that the plural noun, members is not the subject of the verb but is placed in the sentence to distract the unprepared test taker.

This question is horribly written. Who writes like this other than authors of grammatical exercises? The awkward "each of the team members" was obviously concocted to provide a gotcha moment for test takers. And even with all the syntactic gymnastics, the question writer still couldn't get it right. Because this is not a problem of subject-verb agreement, the observation that members isn't the subject (i.e., the head of the subject) is utterly irrelevant. There is no syntactic principle that rules out members as a possible antecedent.

The question of whether a singular or plural pronoun is appropriate really boils down to the semantic question of whether it makes more sense for the plays to belong to the team members, in which case their is more appropriate, or severally to the individual team members, in which case his or her would be defensible. Since team plays typically involve multiple people, I find the first scenario far more plausible, so this question really should be marked "No error".

For a student to make this confusion I can understand. But someone who writes a book about this sort of thing ought to be held to a higher standard.