Question: Why isn’t Simchas Torah celebrated right after Shavuot?
Answer: I guess the reason Shavuos and Simchas Torah are far apart in time is that they are so different from one another. Shavuos is the culmination of Pesach. Rav Shamshon Rafael Hirsch writes that after we were redeemed physically, we were redeemed spiritually as well. G-d chose us as His people, and forged the relationship between us, known as the Torah.
Simchas Torah is the end of a different process. On Sukkos, also called the “Ingathering” of produce in the Torah, we have received the greatest physical wealth of the year. We come to Jerusalmen to thank Hashem (G-d) for it. But that kind of physical wealth requires a spiritual component as well. It begins on Rosh Hashana and Yom Kippur with judgment and forgiveness. On Sukkos and Simchas Torah, we are fully reconciled and close to him again. We are filled with the joy of living a life of Torah together with Hashem. As Rav Hirsch explains again, these are not celebrations of redemption, but of Hashem’s ongoing sustenance.