what's the intuition of the law of large number is important without $iid$ assumption

Question

From the law of large numbers with the $iid$ assumption, the result is that

$n^-1\sum_{i=1}^{n}X_i-E(X_1)I_{\{|X_1|\le n\}}\rightarrow_p0$

or

$n^-1\sum_{i=1}^{n}X_i\rightarrow_{a.s.}E(X_1)$,

It means that we can use the sample mean to estimate the expectation.

But if we only have independent condition, then the result is

$n^{-1}\sum_{i=1}^{n}(X_i-EX_i)\rightarrow_{p}0$ or

$n^{-1}\sum_{i=1}^{n}(X_i-EX_i)\rightarrow_{a.s.}0$

This result shows that we can use sample mean of random variables to estimate the mean of expectation value of the random variables, but I don't know how this result be useful?

In other words, why the law of large number is important, even thouth we don't have the condition that random variables has the same distribution?