Actually we do, technically, and the amount is trivial (and can be reused, though that isn't currently being done). Spent fuel is a political problem. In fact, last time I checked even just the radiation from coal power plants is greater than from nuclear plants.

> The exact same line was fed to us when Chernobyl was state of the art

Chernobyl was never state of the art and western reactors, which are much safer, are also of a design derived from military designs, not optimized for safety. Also, the negative health effects of coal and other energy sources are so much worse than even Chernobyl that it isn't funny.