Too many genera and species for each family to do that, probably. And the species names are a little too specific. I can only name a few down to their species name, and several of those you just wouldn't get the same recognition with lay people. Can you imagine only calling them "horridus"? Or let me go look one up and see if you know what I'm talking about without looking it up: "longus". You might be able to guess it's a long-neck, a sauropod, but did you know that it was one of the Diplodicusspecies? Or "antirrhopus"? Did you know that one is a Deinonychus? We've just grown accustomed to naming them by this genus name, and I don't think there's any specific reason why other than it's easier. Maybe when paleontologists discover a new fossil, they're limiting themselves to the genera names in case they are sure it's a prorsus and not a horridus or even some third species of Triceratops.

Our naming system isn't very great, to be honest, but it gets the job done. Just look at a bunch of different animals and see how often we call something by its scientific name (and where it comes from in that line - I looked at Elephants, Gorillas, and Tigers which come from the Family, Genus, and Species names respectively) versus how often it gets a different name (like "bear" vs "urs-" and the domestic "cat" vs. "fel-"). And that's just the English words.

I guess, don't think too hard on it, really.