What's the usage of $S$ in Dijkstra shortest path algorithm in the book Introduction to Algorithms?
Yes, the set $S$ is not needed in the Dijkstra’s algorithm, which is at the beginning of section 24.3, Dijkstra’s algorithm of the book introduction to algorithms by CLRS, third edition. That is, line 2 and line 6 in procedure DIJKSTRA($G,w,s$) can be removed without affecting the algorithm.

One immediate purpose of $S$ is to facilitate the proof of theorem 24.6 (Correctness of Dijkstra’s algorithm) of the book. Also, other implementations of Dijktra's might use that set more visibly such as the set of visited nodes in Wikipedia's entry. It is indeed a cornerstone concept to understand Dijktra's algorithm.

By the way, your observation that the line 7 of line 7 could be improved to "for each vertex $v \in G.Adj[u]\setminus S$" is very nice. In fact, that is done in the Wikipedia' entry mentioned above. Should that be done, $S$ could not be removed  any more as you have emphasized.