Yeah, I tried that too. I think both eggs & hatchlings count, so you can only get new eggs if you have fewer than 4 eggs and/or hatchlings. I believe adult dragons are unlimited, and if you "freeze" a hatchling it won't count. Also, you can still breed and get new eggs that way.
The "point" is that it is fun and they are cute. Plus it has the pokemon "gotta catch them all" element. I do think they intend to add more varieties of dragons, so that should add to the fun!