In fact, such a great question that the Teimanim (Yemenite Jews) agree with you. They place the beginning of Chamishi right at the perek break. (See here, page 153.) And of course, Stephen Langton, a Christian archbishop of Canterbury, agrees with you, which is why the perek break is right there.
Yet, to explain our standard (Ashkenazi / Sefaradi) aliyah break:
Rony's first answer seems quite compelling to me: "In order to teach us that whoever prays for someone else is responded first." I would not spin it in quite that way, though. The aliyah break, placed later, joins the two sections. And this is because of the logical narrative connection between the women of Avimelech's house becoming pregnant and Sarah becoming pregnant.
The verses at the end of revii are closing up details that were opened earlier. Earlier in the narrative, Hashem had promised the birth of a son, lamoed. And this closes that narrative by detailing the fulfillment of the promise. Then, the detailing of how old Avraham was at a specific time is a way of introducing a new chapter of Avraham's life -- namely, the impact of Yitzchak's birth on his relationship with Sarah, Hagar, and Yishmael. This sort of device might be common at the start of . Consider the start of Toledot, when we are told Yitzchak's age. Or being told Yosef's age at the start of parshat Vayeshev. It is a way of fixing us at a specific point in time, so that we know approximately when to date the narrative that follows.