What I also find interesting is that the nearest extant animals to birds are crocodilians. Both belong to the archosaur clade, even if there's around a 240 million years gap between them.

As you say, birds can be classified as belonging to reptiles under the cladistic route, but they're quite radically different to the reptiles that live today, so are seen as not really reptilian. It's not surprising, seeing that the link between crocodiles, true reptiles in all senses, and birds were the dinosaurs, who disappeared 65 million years ago. A whole lot of evolutionary change in that time.