There's a "cute" series out there somewhere that had Sirius and Remus rescuing a very young Harry and raising him. I think they ended up with like 4 kids to raise actually - Harry, Hermione (somehow?), Draco, and someone else. It was one of those "one big happy family" type stories. Not really my jam but it was one of the better ones of that sub-genre that I'd read, though it did still have a lot of cutesy names like "Pride" and "Pack" for groups of kids and shit. I think it was a series of fics. Trying to find it now.

Ah ha - the Dangerverse. It has a TVTropes page. Here's the author on ffnet. Here's the DLP thread in the Trash Bin - my original rating back in 2010 was a 3/5. I was less picky back then. So take this entire suggestion with a gargantuan grain of salt.

When in Doubt, Obliviate! has Lockhart rescuing Harry. But that's... he doesn't rescue Harry when he's old enough to know what's going on. He takes Harry when he's tiny, if I remember correctly.

So if you are really looking for a story where someone who isn't Snape notices that Harry is being abused and saves him after he's finished with his toddler years... I dunno. Most of this that I've skimmed through has been really cutesy shit. And you're right that it's usually Snape taking Harry in.

Oh, Growing Up Black was decent if I remember right. Try that one.