The use of the 𝑡-distribution and the sample size consideration is not related to the Central Limit Theorem.
Given $Z\sim N(0,1)$ and $V\sim\chi^2_v$ ($v = n-1$ is DOF), if $Z$ and $V$ are independent, then $t=\frac{Z}{\sqrt{V/v}}$.
If $X_1,X_2,\cdots,X_n$ is normal, then$X_1,X_2,\cdots,X_n$$T=\frac{\bar X-\mu}{S / \sqrt n}$.
