Yes, someone else already told me about Shrine Shrine. Of course that was unintentional - it's been fixed and will be corrected with a patch update in the near future.
Pets, like all festivals and most NPC dialogue, are entirely optional. That said, there is a story event that involves the hatchery, and even special skills that you can only see or obtain there.
You can think of it as an unimportant waste of time, but I prefer to think of it as "flavor" for the experience. Really, the whole story is an unimportant waste of time, if you think about it. It's what you put into it that matters and gives it meaning. It's what makes it memorable.
It's like Akamaru says in the manual - if you rush straight to the ending, you're only cheating yourself!