For the purpose of this question, you may assume that we are working over the complex numbers.

Given a connected reductive group $G$, one can choose a maximal torus $T$, and then let $T$ act on the Lie algebra $\mathfrak{g}$ of $G$. One can use this action to define the root datum, which in turn is invariant of the choice of $T$, and use it to classify connected reductive groups.

The action of $T$ on $\mathfrak{g}$ is nice in that it has more information than just the action of $T$ on the Lie algebra $\mathfrak{t}$ of $T$, and yet is simple enough so as to decompose into one dimensional weight spaces. But that is a far cry from saying that this is a natural action to consider when trying to classify connected reductive groups!

Is there a deep reason that root data, or more generally the action of a maximal torus on the Lie algebra of $G$, is a natural thing to consider? Does it correspond to some cohomological invariant? Does it arise naturally? Or is this entire theory a fluke?

The proof does not seem to bring much insight into this story, in that it boils down to a series of reductions, which reinforces for me the suspicion that root data are not in and of themselves natural, but rather that this was a guess for a way to classify connected reductive groups that just happened to work out...