The guard tells you he has been trying to destroy the brotherhood for years. If I happen to run into him while I'm exploring how does it not make sense that he might ask for help? It's make no less sense than someone who has just met you entrusting you with important letters for delivery, etc.
It wouldn't be especially off. It wouldn't be especially great, because I thought initially he was just a bit obsessive. Aren't the DB nearly extinct my character thought?
It would be a viable alternative to killing someone because a child tells you they're a bad person. You're almost making it into a joke to prove I'm not making a viable complaint. It's not a big deal, but why would it hurt if I ran into that guard exploring he mentions what he's doing, or I overhear it and offer to help, like the other 2039820984 quests that can be started that way, and then you get to figure out how to enter the sanctuary, etc? There's no reason that's not viable.
Actually I think it would make much more sense to accept the kids offer, do some snooping to see if he was right, and then have an option on how to proceed. If the bad person is a bad person or not (which I did as best I could with what the game gave me).
Though that last part you could be correct about. It might be me unconsciously applying knowledge to a character that shouldn't have it. I never really thought about how little they'de ever even know about them. But still, it doesn't mean that there can't be alternative ways of accepting the quest. Many other quests are accepting on little to no more knowledge than overhearing someone arguing about it.
I still think a quest that brings them into the light makes the most sense (which we got). If the guard has been hunting them for years and he had the knowledge to take them down (or knew who did) then it would make sense for him to have done it already. If not then you shouldn't be able to magically bring his years long quest to a close. If he wants me to look into it fine, but the DB should be hard to spot. Impossible even. So the neatest solution, to me at least, is to get them to come to you - I liked you weren't entirely in control. You do the kids quest, you still don't need to believe the DB is even around then *whack* your in the shack just rolling with the punches and playing it by ear.