Clades are used for classifying birds and everything else, it’s just a little messier than the neat Kingdom/Phylum/Order/etc way of laying out evolutionary history. If you think of a species as being a specific pinpoint on the tree of life, clades are more like drawing a circle around a lot of pinpoints and branches.

But yeah, biology is nothing but ‘the last thing we taught you was an oversimplification!’ all the way down.