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Thank you for the question. I personally didn't find any contradiction in it, but I thought we could clarify the perspectives involved here...

I think the chapter explains the history of the doctrinal debates by capturing the ways the topics of philosophical inquiry had subsequently arisen and evoked one after another, and by narrating different arguments linearly while moving from one point of view to another--including third person point of view, but obviously not the so-called "omniscient third person" point of view. It thereby offers a holistic outline of the Abhidharma doctrines by meaningfully arranging their connections in a sequence, based on what we can extract from the extant records of the arguments.

When Ch. 4 says, "Then a problem arises", it appears to be a "problem" only in a sense of an issue that had arisen among those who were actively participating in the apologetics and inter-sectarian debates. While it might perhaps be pointed out as a "critical problem" by an opponent party--Vatsiputria in this case,--from the Theravada point of view it may not really be a "problem" in a sense of a doctrinal difficulty per se.

The dictionaries define the word "problem" in different senses:
"a question to be considered, solved, or answered: math problems"
"a situation, matter, or person that is hard to deal with or understand
"a personal matter that causes one difficulty or needs to be dealt with
"a misgiving, objection, or complaint"
"any question or matter involving doubt, uncertainty, or difficulty.
http://www.thefreedictionary.com/problem

The senses fall into different gradations, from the heaviest sense of being the most troubling one causing doubt and uncertainty and threatening the reliability of the whole school, to a light one of merely being a topic of inquiry that awaits to be answered--as in a philosophical problem. It appears to me that the word "problem" here, from the Theravada perspective, can be taken in more or less a light sense of being a new topic of "question to be considered" which had naturally manifested as a result of the process of debates. Perhaps there was a "difficulty" in its task of filling gaps and expressing into a philosophical system with the accompanying terminology and definitions--a task of arranging into more or less a metaphysical or phenomenological system the Buddha himself might have decided to put aside--but was not a reason for "doubt" or "uncertainty".

As a side note, the chapter initially deals with the question of whether or not to accept the "theory of representative perception" in the context of "perceiving agent", of the claim that the "process of perception is a causal process, there is no agent" (p. 4, Ch. 12). But there might be a further problem to "explain the 'continuality', 'samsāra', 'memory' and 'moral responsibility'". The "conceptual process of ten stages" listed here may successfully explain the continuity of the perceptive aspect of personhood--as well as other epistemologically related aspects, including cognition, habits, consciousness, and recollection. An opponent might be inclined to ask for more adequate explanation of the "continuity of rebirth" and "person’s moral responsibility" (p. 3, Ch. 4), although that would be outside of the context of the chapters. An encyclopedia article similarly interprets the motive behind the Pudgala doctrine:

"[T]he Pudgalavādins evidently believed that only such an account could do justice to the Buddha’s moral teaching, to the accepted facts of karma, rebirth and liberation, and to our actual experience of selves and persons.
(Internet Encyclopedia of Philosophy)
http://www.iep.utm.edu/pudgalav/

I happened to agree with you about the soundness of the Theravada doctrine, but an opponent party might have thought of it as a "problem" in an yet another aspect, for it goes on to say: "In their opinions, the dharma theory as represented by the Theravadins led to a complete depersonalization"--in their perspective, that is.

On "Depersonalization" Accusation:
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It is possible that the limitation of available resources we have of Pudgalavada/Vatsiputria gives us, for good or bad, freedom (too much freedom perhaps) and temptation to subjectively read into some aspects of the complexity of what was once an prominent school with a large presence.

"[A]lmost all of the literature of the Pudgalavāda was lost. It is difficult to reconstruct their understanding of the self from the few Chinese translations that have come down to us, and from the summaries of their doctrines and the critiques of their position that have been preserved by other Buddhist schools."
(Internet Encyclopedia of Philosophy)
http://www.iep.utm.edu/pudgalav/

Even while striving to grasp the truth of the matters and maintain a degree of objectivity and academic neutrality, one can perhaps never escape from some degree of perspectival interpretation (inescapable human condition according to Gadamer when understanding other's horizon) and narration ("narrative" in the sense of Ricoeur, not in a sense of fictional story but striving to obtain true meanings), especially when one attempts to render the fragments of history into an intelligible whole.

Some choices of wording could supply meanings to the reasons of why's that would be more familiar to us. When one attempts to attribute implicit motives or read into other's intentions, he might become vulnerable to psychological projection, preference, or influence of academic climate, and it may or may not necessarily be an accurate rendering of what really happened.

For instance, the following statement from another author might be interpreted as being a bit too hasty in projecting some easily recognizable motives to the Pudgalavadins, by taking them to be deliberate "challenge" towards, of all things, the fundamental Buddhist doctrine.

"Pudgalavadins .... deliberately challenged the fundamental dogma of all the other Buddhists. Their motives can be easily understood, for their reaction to the dharma-theory was the same as that of everyone who first hears of it."
(p. 122-123)
"The Personalists represented a reaction against the dogmatic thoroughness with which the Abhidharmists pursued their depersonalizing tendencies"
(p. 124, Buddhist Thought in India: Three Phases of Buddhist Philosophy, by Edward Conze)

Compare with Thich Thien Chau who had written the book, Literature of the Personalists of Early Buddhism, in which he shares a similar but more nuanced interpretation from a point of view of Buddhist adherents, in attributing their motive to instead preserve the fundamental doctrine:

"The creation of the theory of the pudgnla represents a reaction to the depersonalisation of the dogmatic Abhidharma masters. The Personalists (pudgolnvddin), however, were determined to preserve the essence of the doctrine of insubstantiality (anatmavada). They insisted on the fact that adherence to the pudgala did not prevent the attainment of the knowledges (jhana) and fruits (phala). The position of the pudgala was misinterpreted by its adversaries."
(p. ix, The Literature of the Personalists of Early Buddhism)

"The creation of the theory of the pudgala represents a reaction against the "depersonalization" of the dbhidharmika tradition. ...The theory of the pudgala has been misinterpreted by the polemical literature."
(p. 11, "The Literature of the Pudgalavadins", The Journal of the International Association of Buddhist Studies, Vol. 7, No. 1, 1984)

Without dismissing the author's contributions, the choice of certain words shows that it is his own interpretation, especially where he claims that the Pudgalavada was "misinterpreted by the polemical literature" or "adversaries". We obviously need to consider the arguments made by Theravada (Moggaliputta-Tissa in Kathavattu), Madhyamaka (Nagarjuna), and Sarvastivada (or Vasubandhu). I personally hold that these arguments were not "polemical" to be completely dismissed simply as "adversarial misinterpretations"--but of course all of us are entitled to interpretation.

In contrast, the section of chapter appears to me to be more balanced by adding the phrase "in their opinion": "In their opinions, the dharma theory as represented by the Theravādins led to a complete depersonalization of the individual beings". Also, the aforementioned statement from Internet Encyclopedia of Philosophy is careful to specify from which perspective it is "evidently" assumed to be holding: "the Pudgalavādins evidently believed that...".

So far, the word "depersonalization" had been brought up as it were the dominant motive behind the Pudgalavada reaction. Admittedly, sympathetic rendering of Pudgalavada would be more familiar and welcoming to our layman sentiment or the modernity sense of self and individualism. But we perhaps need to keep in our mind too that the notion of "depersonalization" might include some modern or Western senses as well not belonging to the past eras. The Buddha said, "This Dhamma that I have attained is deep, hard to see...", and obviously the doctrine should not be compromised in any way by layman or academic preference.

It is recalled that Nagarjuna also understood the fear that the doctrine of sunyata might invoke:

"Nagarjuna observed a tremendous fear at the heart of our initial view of emptiness.... lurking in the the heart of the experience of emptiness was the terror the possibility of our now not-begin.... This terror could be so great, that, Nagarjuna discovered, one might subconsciously try to hold out something permanent in some conceptual form"
(p. 24, Journeys Into Emptiness: Dōgen, Merton, Jung, and the Quest for Transformation, Robert Jingen Gunn)

"A Modern Reflection on Emptinesss (Sunyata).... Man's ego is alarmed by such a merciless foe, and declares its alarm to be a justifiable 'fear of the void.' ....(Tibetan lama, Tarthang Tulku)"
(p. 76, Understanding the Religions of the World: An Introduction, by Willoughby Deming)