The reason is not in the technology, the reason is in the application.
There are already electron microscopes which have better resolution for much smaller structures. But, for them, you have to kill and fix your specimen. But the hottest area of application of STED is biology. And there, the coolest thing is to get towards imaging in a live cell, so you can observe the biological processes as they happen.
While you could use wavelengths outside of visible light, they disrupt the processes in the living cell a lot. In fact, the intensity of light needed for these super resolution microscopies is so high that phototoxicity becomes quite a concern in everyday research.
And then you also need fluorescent tissue for these methods. We don't have that many ways to produce fluorescent tissue yet, and the ones we have (mainly GFP) have their peaks in the visible spectrum.
The stuff you can do with this technology is amazing. I recently attended a lecture given by Hell (one of the Nobel winners) and he showed a video of recording the electric impulses in the dendritic cells of an anesthesized mouse. It was haunting - beautiful, but also slightly unnerving, to see inside a live brain. This is why we need the light microscopy - our understanding of life itself depends on it.