You know, if anybody except that bunch of brats would actually suggest that the death of Grelod would be a good thing to happen, then I might actually consider working for that goal. But a bunch of kids? There's a reason why underagers aren't allowed to vote or to do important decisions (like: "I'm going to have this medical procedure done to me even though there are risks involved" - the kid is not allowed to decide that. It's the parents or legal guardian who does that).
So, if kids aren't allowed to decide such things, we're going to let one socially peculiar kid decide if another person is allowed to live or to die? I call him "socially peculiar" because I don't think that it's quite "normal" behaviour when someone bangs his head against the floor for months, sitting in a pentagram and trying to conjure a devil or something.
"But she's a horrible old hag!" - yeah, a lot of people are. Do you run around and murder them just because they are?
I think the idea was, you were supposed to pay attention to your surroundings and decide if she deserved to die. Because its fairly obvious she does. (this assumes you remotely consider yourself "good", assassin-for-hire is a valid playstyle in this game)
Correct, traumatized, possibly crazy kid asking for some old woman to be murdered should give a "good" person pause, so start poking around. Walk in (sneak is a good idea, but not necessary) listen to her talking to the kids. Listen to the kids' responses, particularly tone of voice. Check the closet with buckets and shackles on the wall. If that doesn't make you at least consider killing her, you've gone from "good" to "willfully oblivious to evil actions".
Even at that it can be borderline, she is cruel to the point of outright sadism, and clearly a terrible influence on helpless orphans. But does that justify her death? I think barely, but if you really want proof, save your game, kill her, and see what happens. I never got the "yay assassin! I want to be like you!" lines, likely because I sneak and position out of sight when I kill her. I get closer to "I can't believe he actually went through with it!" "Finally!" "Oh, thank you!". Once the other woman calms down (takes days sometimes) she *very* half-heartedly sounds upset at Grelod's death. "I know she was awful, but to be murdered in cold blood? I dunno." That's an anologue to "damning her with faint praise" to me. Yes, there is a standard world-wide guard response of her being murdered, and feeling terrible about it. That's because those are guards who never went to Riften, and just heard of an old ladies' death. There's a Riften-guard specific response of "If you ask me, those brats in the orphanage are better off with Grelod dead." The point is no one in the game world at all thinks her death was a bad thing, and there's good reasons for this. No regrets for the boy who hired you, all the kids are relieved they will suffer less, not the woman who worked with her, not the city guards. You may be right your character can't be absolutely convinced her death is righteous while Grelod is alive, but she absolutely deserved to die.