What makes a bird a bird? - In the Linnean view, at some point, creatures cease being reptiles and evolve into birds. Looking at living organisms, this is a clear-cut distinction. Birds could be distinguished by a long list of list of novel characteristics. When we consider extinct organisms, however, the picture blurs.
Here is a graphic representation of the tree of theropod dinosaur evolution as many biologists envision it, with both modern birds and a representative sampling of non-avian members. These critters pose difficulties for our bird identification scheme.
Alas, we can't. No uniform criterion exists for choosing key characteristics in such circumstances. Consequently, whichever characteristic we pick, be it furculae, feathers, or toothless beaks, must inevitably be arbitrary and subjective, in other words, unscientific. Better to base our definitions on something more concrete - the fact that birds are descended from the common ancestor of reptiles. To say that birds are members of Reptilia reflects this pattern of descent without invoking arbitrary criteria.
Situations in which one group disappears by evolving into something different is termed a pseudoextinction. As a practical matter, if we wish to count either extinctions or numbers of Linnean taxonomic groups we must somehow factor these out.
By the way, if you don't buy the dinosaurian origin of birds, it really doesn't matter. Regardless of which reptiles one thinks birds descend from, one faces analogous problems.