This one's a tough one, kudos to you for being thoughtful before possibly committing 10+ years to taking care of a dog.
Kids will be kids, and I don't think they have any idea of what it takes to care for a dog. Maybe, and that's a very non-committal maybe, your 15-yo might have it in him, but chances are it's going to fall to the adults in the family. IMHO, if you're against it, and your wife is only "leaning" towards it, it's a bad idea.
Letting your kids have a trial is a good idea. Since they're the ones who want it, they're the ones who have to prove to you that they can do it. I don't know your kids, but I really doubt the 10-yo has what it takes.
A dog is a part of the family and a huge commitment, I would go so far to say that I would think about it as hard as I would about having a child. It would be a shame for it to be brought home as a toy, to be eventually marginalized and then discarded. It happens far too often, and I really, really hope you make the right choice.
Good luck, and go Niners.