Why is standard error of the mean always calculated from the population variance? The standard error of the mean is often calculated with the formula $\sqrt{\sigma^2 / n}$, which uses population variance (or an estimate of it).
However, standard error of the mean is defined as the standard deviation of the sample means. So why don't we ever directly calculate standard error of the mean as $\sqrt{E[(X_i - \overline X_n)^2]}$ where $X_i$ is a sample mean and $\mu$ is the mean of the sample means? Since the formula for standard deviation is $\sqrt{E[(X - \mu)^2]}$.

The "standard error" of an estimator is just the standard deviation of that estimator.  Typically this depends on the unknown parameters in the model, and so the true standard error is a function of the parameters and is unknown.  We can then estimate the true standard error to get the "estimated standard error" which is now a statistic that depends only on the observed data.  In the context of using a sample mean as an estimator for the population mean, we have:
$$\begin{align}
\text{True standard error} 
&= \mathbb{S}(\bar{X}_n) = \frac{\sigma}{\sqrt{n}}, \\[12pt]
\text{Estimated standard error} 
&= \hat{\mathbb{S}}(\bar{X}_n) = \frac{s_n}{\sqrt{n}}. \\[6pt]
\end{align}$$
As you can see, the estimated standard error in this problem does indeed use the sample standard deviation instead of the true standard deviation in the model (though it is still different to what you're proposing, which does not seem to apply the correct scaling at all).
