"Under that definition, we would have to carve up dogs into several different species, as many breeds of dog cannot naturally mate with other breeds (Great Danes and chihuahuas)."
Not only that, but what about two breeds of dog that can only produce fertile offspring 50% of the time? 90%? 10%?
The black and white areas are easy- housecats and tigers are clearly different species. The gray areas are a bit tougher- lions and tigers can interbreed to create fertile offspring. Perhaps with several more million years of speciation, lions and tigers will lose this ability to interbreed.
Species aren't distinct pigeonholes, but rather a continuum with limitless gradations along the way.