According to what you write, X is normally distributed with unknown mean and unknown standard deviation. The true unknown mean is mu, and the standard deviation is t. Then (x-mu)/t would be a standard normal random variable (not (mu-x)/t). The maximum likelihood estimator of mu would be the sample mean of the S draws, and the maximum likelihood estimator of sigma would be the square root of the sample variance, after scaling by (S-1)/S. You can make these estimators as precise as you like by increasing S.
Now, if X is not actually normally distributed, then you would need to call on the LLN and the CLT to get to an asymptotic normal distribution. But, you could still estimate mu and sigma in the same way, they just wouldn’t be ML estimators, they would be method of moments estimators.
Actually, if you can sample the R.V., you must know mu and t, no? So, I must be misinterpreting your question.