Question: Since Shavuout celebrates the giving of the Law, why isn’t the cycle of Torah celebrated at Shavuot, rather than at Simchat Torah (the special holiday at the culmination of the Succot holiday)?
Answer: Thank you for your question. It is an excellent question, which deserves a much longer answer. Basically Shavuot is the anniversary of giving the first tablets. These tablets were destroyed by Moshe 40 days later when he saw the Golden Calf, so we never actually received them in our possession. Shavuot is therefore more about a relationship with G-d than the Torah itself. This is why we have the custom to stay up all night, even though we will learn less Torah, we show our close relationship to the One who gave it.
The Torah was actually given to us on Yom Kippur, 120 days after Shavuot. Because of technical reasons of too many holidays with other themes, we delay our celebration of the Torah itself until Simchat Torah. That is the day that G-d gave it to us to learn and develop. I hope this answer helps
All the Best,
Rabbi David Sedley