hypothesis test for a mean: why use sample std dev ?

When a simple null hypothesis: mean = mu0 is made, and we want to test it against some alternative, all stat references I can find compute (x_bar - mu0)/(s/sqrt(n)), and find the P-value using the t-distribution with n-1 dof. Here s is the sample standard deviation, computed using x_bar. Since we are assuming the mean to be mu0, why don't we use that information when computing the variance of the sample, rather than x_bar ? I realize this would have a different distribution than T(n-1), but wouldn't it be more accurate to figure out what that distribution is, and use it instead ? Thank you.