"As the crow flies" is still used frequently in my corner of the USA. And it really does mean, as a bird would fly it, not by the roads or from a map.
A Southerner might say, "Well, it's ten miles by the road, five as the crow flies." Both are equally correct. In terms of basic distance, the destination may just be five miles, but there are ten miles of roads leading to it.
Like many idioms used frequently in the South, this is also of English/Scottish extraction. Most of the more colorful expressions used in this part of the country either have a definite Celtic-Anglo origin, or they come from the African-Americans who derived the expressions from their ancestors.