Autocorrelation for Stationary Signals
You need the joint distribution of $X_{t_1}$ and $X_{t_2}$ in order to calculate $E[X_{t_1}X_{t_2}]$ and nobody has claimed that the joint distribution of $X_{t_1}$ and $X_{t_2}$ is the same as the joint distribution of $X_{t_1}$ and $X_{t_3}$. Stationarity requires that the joint distribution of $X_{t_1}$ and $X_{t_2}$ be the same as the joint distribution of $X_{t_1+\tau}$ and $X_{t_2+\tau}$ which is a quite different requirement than the one you need for claiming that the autocorrelation function of the process is a constant.  See this answer of mine for more than you might ever have wanted to know about autocorrelation functions.