> [Birds] are ... characterised by feathers, toothless beaked jaws, the laying of hard-shelled eggs, a high metabolic rate, a four-chambered heart, and a strong yet lightweight skeleton.
Notice that when you put all of those traits together, you get the ability to lead a lifestyle centered around long-range migratory flight.
One or a few species evolved this particular set of traits, which gave them the ability to forage incredibly long ranges in search of food, in a landscape that was—at the scales other dinosaurs could travel—increasingly inhospitable to life. This small set of species then took over the habitats of every other dinosaur species as they died out, and then speciated into all modern birds. It was a sort of evolutionary bottleneck point.
In other words, "birds" are essentially defined as the set of theropod species that survived the K-T extinction. Birds happen to all have many unique traits in common, because these traits were adaptive through the rapid change of environment that killed off the other dinosaurs.