Does every smooth local frame of the tangent bundle correspond to a chart?

You are right. If you want the statement is true around a neighborhood you should look at @Neal answer instead of mine in that post, which more or less refer to the following theorem.

The necessary and sufficient condition for a smooth frame $\{X_i\}$ to be expressible as a coordinate frame in some smooth chart is that they are a $\textbf{commuting frame}$, i.e. $[X_i,X_j] = 0$ for all $i$ and $j$. This theorem proved in John Lee's $\textit{Introduction to Smooth Manifolds }$ book at Theorem 18.6 here.

So not all local frame expressible as coordinate frame. Only the commuting ones.