Personally, I like to think about this every time I see a pigeon. Sure, there are more dinosaur-like birds out there -- Hanson uses the example of the cassowary, which is indeed very much like something you'd see in Jurassic Park -- but you can spot the dinosaur lineage in any bird if you're looking closely enough. For one example, check out the video around the three-minute mark, where you can see footage from an experiment that gave birds tail prosthetics to make them walk with old-school dino swagger.
Of course, not all dinosaurs had the privilege of being the ancestors of the brood that lives today. All of the species that we consider dinosaurs are long extinct, and plenty of lineages went extinct, too. One branch of the dinosaur family produced the oldest common ancestor of all living birds. We still don't know exactly when that happened, or where we can draw the line between "bird" and "dinosaur." But we know that plenty of dinosaurs were quite unrelated to that proto-bird, and they've been lost from the family tree of life.
At the end of the video, Hanson comes to one of my favorite things about bird evolution: Birds survived while other dinosaurs perished because they got super tiny super fast. In a world of towering giants, some dinosaur groups shrank to fill a different evolutionary niche. In other words, there was too much competition among bruisers who went after large prey and grazed on tall trees. These tiny dinosaurs were able to seek out other resources -- animals too small to interest their cousins, and plants too tiny for them to get to easily.
When the dinosaurs started to die out, it was the little guys who were still able to find the food and shelter they needed to survive. Ergo birds.
So if Jurassic Park was a real-world project, the first resurrected dinosaur on display probably wouldn't be a giant T. rex. It would probably be some sort of bird that had been modified to resemble one of its ancient ancestors. Think a chicken that's been spliced to fulfill its dino potential sounds far-fetched? It might be possible right now, thanks to a gene editing technique called CRISPR. Listen to the latest episode of Radiolab for more info on our future dino-chicken overlords.