Simchat Torah, celebrating the completion of the public reading of the Torah, occurs at the end of Sukkot, apparently for one or more of the following reasons.
1. We celebrate the most happy occasion of finishing the reading of the Torah during the holiday on which happiness is most stressed by the Torah, namely, Sukkot.[1]
2. The reading of V’Zot HaBeracha, Moshe’s blessing to Bnei Yisrael, at the conclusion of the Torah, is appropriate at this time, as it corresponds to Shlomo HaMelech’s blessings to the nation at the end of Sukkot after the inauguration of the Beit HaMikdash.[2]
3. Another parallel is that Simchat Torah is the last day of the shalosh regalim of the year.[3] Rav S.R. Hirsch explains in Horeb that the specific significance of Shemini Atzeret/ Simchat Torah is to “gather in” the spiritual lessons of the holidays and to concentrate on having them last through the long winter. This idea could also apply to the completion of the Torah cycle.
Regarding your suggestion that we finish and restart the Torah reading on Shavuot, consider the following. The Torah was given on Shavuot, but not specifically the text of the Torah that we complete on Simchat Torah. There are two opinions in the Talmud[4] concerning when the Torah was written down. One opinion holds that Moshe wrote it in sections, as matters transpired, and completed it directly prior to his death. The other opinion holds that the entire Torah was written down at the time soon before Moshe’s death. Certainly, it was not presented in its current written form at Har Sinai. At Har Sinai, on or around the time of Shavuot, HaShem gave most of the laws of the Torah with their explanations and halachic implications along with the principles of the Torah – not the exact text of the Chumash. Therefore, Shavuot need not be the right time to finish the Torah cycle.