Yeah, kookaburras are one of those birds that are great for flying and keeping in an aviary, but don't make great pets. They're pretty snuggly birds, but act a lot wilder than most parrots, I find.
I've only ever seen them kept as pets in America, in Australia you're allowed to own them but no one really does.
Like you mentioned diet is a big factor, but then there's also the fact that - like toucans - they can't climb with their beak like parrots, so they need to be in an aviary big enough for them to fly around in.