A simple proof would be like
$$P(X < x|Z = 0) = P(T < x) = P(S < x) = P(X < x|Z = 1)$$
So knowing the value of Z would not change the conditional distribution of X, hence independence.
Now back to your question. You make mistake about the concept of X and S. X is the smaller value between S and T, but knowing the value of X does not mean you will know of which this value comes from, S or T. So when you say
X = min(S, T) =S
you are actually conditional on both the value of X and S, not just X, then Z would of couse be dependent on both X and S.
But Z and X are independent.