It is unusual to have Torah reading during the afternoon prayers and particularly uncommon to have it include a portion from the prophets as a Haftorah. It only occurs on fast days, not even on the Sabbath The haftorah read, including on Tisha Bav, is always the same, Dirshu from Isaiah fifty five. It is ultimate expression of repentance leading to the ultimate redemption. The initial verse hints at the period between Rosh Hashana and Yom Hakippurim, which should have led to it being read on Yom Hakippurim itself.
Strengthening this presumption is the view of the Mechaber, that after reading the Haftorah during Mincha that the blessings recited don’t mention that the day is Yom Kippurim indicating that the reading reflects that it is a fast day. Even the contrasting view of the RAMA which requires adding a blessing on the sanctity of the day doesn’t exclude the likelihood that the reading reflects the aspect of the day’s character that reflects it being a day of fasting.
The question becomes why we don’t read that portion and instead read the book of Yonah?
The fact that on Yom Hakippurim there is a different Torah reading doesn’t help to answer the question since there is no clear connection between the Torah reading, which focuses on the sin of illicit relations, and anything mentioned in the book of Yonah.
Both choices speak about repentance and how it transforms. What is unique about Yonah is that instead of a theoretical discussion of the benefits of repentance it describes a real event. The prophet was unwilling to fulfill his mission because he feared success. The people of Nineveh, who will eventually defeat and exile the Jewish people, would repent and will be saved. Irrespective of would later occur the people did repent at that time and were saved.
What differentiates Yom Hakippurim from other fast days is that repentance on this day leads to forgiveness and renewal. If the non-Jewish people of Nineveh are forgiven and saved, then our fasting and repenting should certainly lead to our being forgiven.
After a long and arduous day of fasting, praying and examining our sins, it is highly probable that we would become discouraged as we honestly acknowledge our weaknesses and there is a need for reassurance. Previous years have demonstrated that no matter how sincere we are during Yom Hakippurim and how highly we are spiritually elevated the impact over time fades. We need a reminder that as long as we are committed on this day Hashem will purify us and grant us atonement.
Every year we need a new Yom Hakippurim because we are human and fallible. However, we are capable of changing and growing and the unique character of the day enables us to fully repent. If the city of Nineveh which doesn’t have our special relationship with Hashem can repent and be saved, then we certainly are capable of doing at least the same. Yom Hakippurim is demanding as well as uplifting, but we can succeed and be redeemed and transformed.