Generating set of a group and relation to diameter? Is there an efficient algorithm to test if a given subset of symmetric group generates the symmetric group?
Drawing the Cayley graph and testing for all pairs reachability is one way however that scales in the size of the group.
Is there a relation between maximum distance between generating elements where distance betwee two elements of generating elements is with respect to transpositions and diameter of Cayley graph of the group with respect to the generating set?
The answer to the first part of your question is yes.

Given a permutation group $G = \langle S \rangle$, there is a polynomial time algorithm (clearly more efficient than the brute force approach which has factorial complexity), the Schreier-Sims algorithm (https://en.wikipedia.org/wiki/Schreier%E2%80%93Sims_algorithm) which computes a base and strong generating set (BSGS) that can be used to compute the order of the group (hence decide if $G=S_n$).
A good reference seems to be Seress' Permutation Group Algorithms and especially chapter 4.

Also, this question can be solved in practice by using GAP $\texttt{IsSymmetricGroup}$ (https://www.gap-system.org/Manuals/doc/ref/chap43.html), which implements this method.