Fantastic question. I haven't read The Closing of the Western Mind, so I'm basing this answer off of your summary. It's not quite accurate to call Egypt a personal province. It was an imperial province, which is a subtle but important distinction.

The topic of private property in the ancient world is kind of complicated, because as a concept it doesn't always directly translate. This is especially true when thinking about the possessions of rulers, whose incomes can't always be neatly separated from those of the state. The line between the public and the private is one issue that has caused a lot of debate among scholars regarding the economy of Ptolemaic and Roman Egypt, however we can probably gloss over that for now.

Historians have put a lot of effort into studying the immediate aftermath of Augustus's conquest of Egypt, because the way it was integrated into the Roman Empire was in some ways novel. Dominic Rathbone in his book Egypt, Augustus, and Roman Taxation calls the province a "laboratory" for Augustus to test his approach to taxing and administering Roman territories.

Ptolemaic Egypt had a fairly complex, if at times dysfunctional, administration by the 1st Century BCE. Rather than uproot this system and have to start from scratch, Augustus retained as much of the old government as possible. The old layers of government, from the provincial level - Egypt was traditionally divided into 42 nomes or provinces - down to the municipal level, were left mostly intact. If the state in early Roman Egypt functioned at times like an estate, this was a legacy of the Ptolemaic period.

Above all of these offices, Augustus appointed a prefect, essentially a governor, who answered directly to the emperor. These men were selected from the equites rather than the senatorial class, and were appointed directly by the emperor, who could (and did) replace them at will. Unlike proconsuls, the prefect did not have a predetermined term limit, so his length of service was limited only by his usefulness.

This was what made Egypt an imperial province, differentiated from so-called senatorial provinces. For the latter category, the Senate was at least nominally in charge of appointing proconsuls to govern them, though in practice, Augustus held considerable personal sway over the administration of the empire. Egypt was the first "imperial province" to be governed in this way, but afterwards many border provinces were governed in a similar fashion. That new administrative structure was one of the more successful ideas to emerge out of Augustus' policy in Egypt, if we define "successful" as long lasting and widely applied.

There are a lot of reasons why Augustus might have done this, some fiscal and some political. Broadly speaking, Augustus made a policy of appointing officials in Egypt solely from the equites, rather than the senatorial class. This was a way for him to exclude the senatorial class and wield more autocratic power in Egypt. By appointing a member of the equestrian class whose authority solely derived from his role as an instrument of the emperor, Augustus maintained more direct control over the province.

It wasn't a perfect system, because it essentially amounted to selecting someone he trusted, preferably with military and political experience, and sending him to the far end of the empire with 3 legions under his command. In Egypt, the prefect largely filled the role of the emperor in lieu of his presence, passing laws and putting down rebellions when they occurred. Left in charge of an enormously populated province with massive revenue, it was the ideal place for an ambitious man to try to seize more power for himself. More importantly, it was a situation in which many emperors, including Augustus and his successors, might feel insecure in their prefect's loyalty. Cornelius Gallus, the first prefect appointed by Augustus, was disgraced and fell out of favor with the emperor because he overstepped his station. His tenure lasted just 4 years, and most of his successors did not fare much better. Domestic uprisings and hostile neighbors led to embarrassing defeats for many prefects, and still others lost their positions for more personal reasons.

When Augustus conquered Egypt he naturally confiscated all of the estates previously owned by the Ptolemaic monarchy, and redefined them as public land (ager publicus). All of the rent, taxes, and revenue generated by public lands went back into the imperial government, particularly the provincial government, and in that way indirectly benefited the emperor himself. Land was also confiscated from some Egyptian temples and added to the public land. Private land was usually not confiscated, except if the previous owner had done something to warrant punishment. Under this system, many people lived and worked on "public" land. Some of these estates included in the category of public land were personally owned by the emperor himself, his family, or his friends. However this dynamic was not unique to Egypt, nor is it always straightforward to describe these situations as outright privatization.

Feel free to ask any follow up questions, this is a rough generalization of many overlapping developments which occurred in Egypt after the Roman conquest. Someone more familiar with administration in other Roman provinces might be able to weigh in on taxation and imperial revenue during the Late Augustan era, and to what extent Augustus personally profited from taxation.

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