Given a simple complex Lie algebra $\mathfrak{g}$, recall the Springer resolution of its nilpotent cone $\widetilde{\mathcal{N}}\to \mathcal{N}$. Several times I have seen someone explaining Springer theory in terms of perverse sheaves, and each time the existence of the Grothendieck-Springer alteration $\pi: \widetilde{\mathfrak{g}}\to\mathfrak{g}$ along with the diagram $$\require{AMScd}\begin{CD} \widetilde{\mathcal{N}} @>>> \widetilde{\mathfrak{g}} @>>>\mathfrak{t}\\ @VVV @VVV @VVV\\ \mathcal{N} @>>> \mathfrak{g}@>>>\mathfrak{t}/W \end{CD}$$ is magically pulled out of a hat (eg: There also exist this other thing that...), where $\mathfrak{t}$ is the universal Cartan and $W$ is the Weyl group. Then one uses the fact that $\pi$ is a small map, giving an IC sheaf $\pi_\ast\underline{\mathbb{Q}}_\widetilde{\mathfrak{g}}$ with a natural $W$-action, which in turn induces a $W$-action on the Springer sheaf by some functoriality. I find this unsatisfying because it seems like $\widetilde{\mathfrak{g}}$ is kept mysterious.

My vague question is how should I think about the Grothendieck-Springer resolution and what is its role in modern representation theory? I know this is not a good question, so let me try to refine it by asking the two following questions.

1) Is there a broader theoretical context to fit the above diagram into where I am given a resolution of singularities $X_0\to Y_0$ (maybe with $X$ symplectic?), and can find a smooth family $X\to T$ and a proper map of smooth varieties $X\to Y$ fitting into the diagram $$\require{AMScd}\begin{CD} X_0 @>>> X \\ @VVV @VVV\\ Y_0 @>>> Y, \end{CD}$$ or is the Springer map special in a sense I don't understand?

Aside from applications to proving a generalized Springer correspondence, are there other examples where the existence and properties of this remarkable space are used in representation theory?

2) What are other applications of the Grothendieck-Springer resolution?

For example, the Springer resolution can be interpreted as a moment map, and David Ben-Zvi's answer to this question shows how this may be interpreted as the semiclassical shadow to Beilinson-Bernstein localization. Is there an analogous quantization of $\widetilde{\mathfrak{g}}\to \mathfrak{g}$? EDIT: I would be particularly interested in applications which are not so closely connected with the Springer resolution.

I'll stop here, since I have probably already asked too many questions. I would greatly appreciate any references to a modern understanding of $\widetilde{\mathfrak{g}}$.