Either way its debatable if pet battles has no correlation to the game itself . People use pets to fight in real life for entertainment. Our characters can do that too.
Who are you to decide what goes in a game or not?
They decided to put it in. Some people enjoy it, some peole dont, like any other content. But unlike other content (Like raiding) It does what its supposed to do pretty well.
I LOVE raiding in MoP. But they are having trouble trying to figure out how difficult a raid should be (LFR/Normal/Heroic). People argue almost everyday if normal is undertuned/overtuned and if the nerfs were warranted or not. The appeal to raiding is to too many people and they haven't hit the sweet spot yet. (Not sure if its possible)
Same thing for PvP. Notice how much its been changed just this xpac alone (in terms of gearing) I suspect its because they aren't happy the way they are/were and are trying to 'fix' it to make it better.