Exercise 5 (Vinogradov-Korobov bound) in Tao's Math 254A Notes 5 (Bounding exponential sums and the zeta function)

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In Terry Tao's Math 254A Notes 5 (Bounding exponential sums and the zeta function), Exercise 5 outlines the derivation of the Vinogradov-Korobov bound for Dirichlet $L$-functions. Let $\chi$ be a non-principal character of modulus $q$. The goal of part i is to show:

$$L(\sigma + it, \chi) \ll \log^{O(1)} (q|t|)$$

whenever $|t| \ge 100$ and $$\sigma \ge 1 - O\left( \min\left( \frac{\log\log(q|t|)}{\log q}, \frac{(\log\log(q|t|))^{2/3}}{\log^{2/3} |t|} \right) \right).$$

While a rough outline can be structuralized by adapting Tao's analogue proof for the Riemann $\zeta$-function, the finer analytic machinery remains highly implicit. In particular, propagating the Tao-style smooth approximation to Dirichlet $L$-functions, rigorously controlling the error terms via Poisson summation, and optimizing the phase scale parameters via Vinogradov's estimate:

$$\frac{1}{N} \sum_{n \in I} \e{f(n)} \ll N^{-\frac{c}{k^2}}, \quad \forall T \le N^k$$

(under the regime $2 \le N \ll T$, $|I| < N$, and $|f^{(j)}(x)| \asymp \e{O(j^2)}\frac{T}{N^j}$) requires a non-trivial alignment of local estimates.

Could someone provide a completely self-contained, rigorous derivation of this bound (Part i) that fills in these analytic gaps?