"The problem is the somewhat sloppy definition of a species."
The problem is that the idea of a species is almost prescientific and arose before modern genetics. To use an analogy which may be illuminating but is probably totally wrong, it's like someone attempting to characterise cars without (a) knowing about manufacturers and (b) knowing about engines and transmissions. And then, when it turned out that the original classification system wasn't based on the deepest underlying facts, trying to keep it going regardless while shoehorning in new understanding.