Kervaire's seminal 1960 paper A manifold which does not admit any Differentiable Structure starts "An example of a triangulable closed manifold $M_0$ of dimension 10 will be constructed."
What is the importance of triangulability of a manifold? It suggests a certain degree of "niceness" but what is gained by this assumption precisely?
I don't think you should view a triangulation as a "niceness" feature of a manifold. Keep in mind that this paper was one in a chain where people were answering various foundational issues that came-up when dealing with the concept of a manifold.
It goes back to Poincare's work. His proof of the duality theorem for manifolds assumes manifolds are triangulated. In particular, his definition of homology required a triangulation. So a natural question people came to ask was if every manifold has a triangulation, and whether or not they're essentially unique (up to subdivisions). Other side-questions were whether or not you could define homology without triangulations, in a homotopy-invariant way, etc.
Poincare assumed manifolds were smooth, and at that level of generality the question was answered by Whitehead: smooth manifolds admit unique smoothly-compatible triangulations up to subdivision. But topologists eventually took this foundational problem in a more general light. With the development of point-set topology one can ask about triangulations of topological manifolds. Here things become more subtle and technical, as it is asking for a feature which is "native" to smooth manifolds (triangulations) to apply to a very different kind of object, a non-smoothable manifold.
So this is much like trying to find analogues of ideas from Riemann manifolds for plain metric spaces (say, metric spaces with path-metrics).