Why does the Bible say there are different races of people...

[Cato]

And he was dead wrong - there's no such thing as race.

You're being idealistically absurd. I agree with you, but only in the context of a strict taxonomical definition of the word. This consideration doesn't mean that long standing traditional understanding and use of the word suddenly become invalid. Even if we were to adopt a strict taxonomical definition of race we would have to come up with another term to take the place of its more conventional use. Perhaps you would feel more comfortable using breeds instead, like we do for Canis lupus familiaris.

If the concepts behind race, racism, and race relations make you uncomfortable then you're one of the good guys. Enforcing a strict use of the word race will not eliminate discrimination based on superficial biological or cultural differences. A campaign of this nature is quixotic.

The all too human fear of 'the other' is at the root of the common understanding of racism, not a belief that others are of a different species. We have made significant process regarding racism. The fact that it didn't immediately disappear with the knowledge that we are all the same species should give you pause for thought.