Why are viscosity solutions to Hamilton-Jacobi equations interesting?

The method of vanishing viscosity is not a particularly good way to think about viscosity solutions. The reason viscosity solutions are so useful is that they satisfy a maximum principle (or rather a comparison principle), which gives them very nice uniqueness, existence, and stability results (making it easy to pass to limits, for instance). For these reasons, viscosity solutions are almost always the correct notion of solution in applications, which include things like optimal control theory and geometric flows like mean curvature motion, to name a few.

Viscosity solutions apply to fully nonlinear equations for which no other notion of solution is available (e.g., classical solutions do not exist, and weak distributional solutions do not make sense). Without viscosity solutions, we would have no way to rigorously make sense of a very broad class of PDEs. I could go on, but others have written more elegant answers to this question elsewhere:

https://mathoverflow.net/questions/59449/why-are-viscosity-solutions-useful-solutions