Aside for Mordechai’s desire to send these letters in Sivan for the reasons mentioned above, the Yosef Lekach writes that Mordechai waited for Haman’s couriers to return from their original mission (Esther 3:13). Utilizing the same couriers would add legitimacy to Mordechai’s letter.
The Midrash (Bireishis Rabba 100:6) writes that H-Shem rewarded the gentiles for mourning Yaakov for 70 days (Bireishis 50:3) by giving them these 70 days between the 13th of Nisan and the 23rd of Sivan to do teshuva.
R’ Yehonason Eibshutz writes that a common calculation in the Torah is a day for a year, as when the Jewish people were punished (Bamidbar 14:17) with 40 years of delaying their entry into the Holy Land for their believing the spies who traversed the land for 40 days. Therefore, he writes, these 70 days were for the Jews to perform teshuva in gratitude for H-Shem’s saving their lives, which the verse (Tehillim 90:10) says lasts an average of 70 years.
Similarly, the Vilna Gaon explains that the Jews were scared about their fate for these 70 days to get an atonement for the 70 years of exile which they had caused upon themselves.
R’ Dovid Feinstein writes that these 70 days represented the 70 nations of the world were allowed to think that they were in control of the fate of the Jews.