Edit: so i don't remember much about Naruto lore, but i think he lived in an orphanage for a while then was moved to his own apartment building. In this AU it was that but then moved into Sakura's home. Does Naruto have any positive or negative feelings about his orphanage?
Gonna be honest, I'm not 100% on Naruto's canon childhood. They were kinda vague and just threw some random ideas out later and I didn't like read the data books or anything.
I am sure he sat on a swing though, mmhmm, definitely at least looked at a swing.
In this story, he was secretly housed until he started going to the Academy, which is when him coming back to any of the important ninja tossing him around like a hot potato would suddenly be noticed, because he finally had to leave the house.
The goal really was to avoid anyone finding out he was that important, and there's not a ton of risk of that if no one knew where he was because he never left, so holding him between locations and transferring him with ninja precision was fine. You can safely assume pretty much everyone that should have been looking out for Naruto, babysat at least a little bit during this period. There's not a whole lot of explanation otherwise for how 3 year old Naruto survived besides an orphanage, and I just don't think any of the war-hero, panic prone, PTSD for breakfast ninjas would have been fine with him in an orphanage. Just not buying it, I worry someone's gonna break into my house and steal my PC every time I leave my house and that was just money, and I definitely didn't go through the fuckin PC wars.
After the Academy started, and he needed to get back and forth without causing problems, they set him up in his own apartment in a medium quality district near the Academy, just called it good. Left him with an okay amount of money per month, nothing crazy. When 6-7 year olds start doing Itachi things sometimes, and you're from a generation of war and nonsense, it can be hard to remember that no, 6-7 year olds should not be forced to budget, but I actually think that's canon so it works here.
At around 10 actually valid complaints were made, because Naruto was really fucking loud. Turns out excited, training kids that don't super have to sleep are obnoxious.
And stepping in to tell everyone to shut the fuck up as a ninja order from either Kakashi, Jiraiya, or the fucking Hokage, would make his importance really obvious.
So he was simply moved elsewhere, where he learned to be quieter.
And then Sakura showed up and was like "The fug? Just come live with me, why weren't you already living with me? It's weird that you weren't living with me."
And the Hokage and co were like "why didn't we think of that?"
Used some Anbu to watch the house to ensure he wouldn't be mistreated for like a week, then called it good.
That's also why Naruto's lock works, by the way, that was an earlier joke but for real the Anbu switched it out and sent the Haruno's a bill and Mebuki was like "We hadn't fixed that damn lock? Huh."