Well, from these pictures, I think you can see that there is in fact a visible ice sheet at the poles, and you can see that global dust storms do in fact kick dust over the poles. Dust that blows into the polar region can also blow out of the polar region, but surely some of it gets stuck there. So how do the poles stay white?
The main reason is that the polar ice evaporates every summer and re-deposits every winter. The shiny white ice isn't just the same ice year to year, it's constantly melting out (or sublimating, really) and being renewed, transferring dust to the ground underneath, so what you see is always nice and shiny white.
Hubble got some nice photos in the mid-90s to show how dramatic the seasonal difference can be:
Even at the most extreme latitudes where the ice never fully melts, it thins in summer and thickens up in winter as the atmosphere freezes and deposits, and dust will accelerate the melting process. A patch of dust at the pole is less reflective than the ice around it, so it will heat up and melt its way below the surface level.
In large part, this is all very similar to how it works on Earth. When the sun comes out after a snowfall, if you place a dark object on the snow, it will rapidly melt a hole as it sinks down to grass level. Our own poles and permanently white mountaintops must be subject to a certain amount of dust and debris blowing in, yet they're always bright and shiny century after century. Surface ice comes and goes.