Estimation of std error using sample std dev. will result in diff. sample statistic distribution for diff. samples. Won't this affect our decision?

Question

During hypothesis testing (say, comparison of sample mean to a hypothesised population mean), we calculate p-value using sample mean distribution. For constructing this sample mean distribution, we need:

1. mean of the sample mean's (or mean of the population): This we get from our hypothesis.
2. std error of the sample mean's: This equals to population std dev/sqrt(sample size) . Since we do not know the population std dev, we use the sample std dev. as an estimate of population std dev.

Now, the doubt is, for different samples, we will have different sample std dev's, so we will end up with different sample mean distributions. Hence the p-value shouldn't be necessarily comparable. Can someone please help and explain what I am missing here?