I think we all sometimes need to remember that Sirius spent his time in azkaban surrounded by literal depression demons and he did not have a good time there.
Lmao my favorite thing I hear Remus apologists say is "Well Sirius didn't break out of Azkaban until Harry was 13, so he clearly didn't care about Harry, how dare you call Remus out for not checking up on Harry in all that time! What Sirius did was just as bad, if not worse!" Classic.
Sirius Black to Harry in PoA:
“Harry ... I as good as killed them,” he croaked. “I persuaded Lily and James to change to Peter at the last moment, persuaded them to use him as Secret- Keeper instead of me. ... I’m to blame, I know it. ... I set out for your parents’ house straight away. And when I saw their house, destroyed, and their bodies ... I realized what Peter must’ve done ... what I’d done. ...”
In the book:
“You killed my parents,” said Harry, his voice shaking slightly but his wand quite steady.
Black stared up at him out of those sunken eyes.
“I don’t deny it,” he said very quietly. “But if you knew the whole story.”
In other words:
Sirius blames himself for James and Lily’s deaths, even though he knows he wasn’t the traitor. That’s why he doesn’t try to escape Azkaban for the first 12 years - he believes he DESERVED TO BE THERE. He believes that he does not deserve to look after Harry, after all, he’s the reason they’re dead!
Knowing he wasn’t the traitor (even if he deserved punishment for his direct role in causing their deaths) and being able to turn into Padfoot kept him sane in Azkaban. It is only when he looks at the newspaper and sees Peter Pettigrew staring back at him as Wormy - with his missing finger - that he realises that Peter is in Hogwarts, and Harry is in danger.
Up until then, nobody, not a single prisoner, had ever escaped Azkaban without help. Yet within a short while of Sirius finding out Harry is in danger, he managed not only to come up with an escape plan, but to execute it - with no help - flawlessly.
And then - he does everything in his power to help and protect Harry, not only risking his life (and ultimately loosing it) but also risking a fate worse then death - the dementors’ kiss - in doing so. And, he offers Harry a home and a promise that they’ll be “a proper family when this is over”…
I do think Remus has reasons (aka stupid “too poor, too dangerous” type shite, some of which is actually very important) for not seeing Harry or trying to help, but compared to Sirius his excuses aren’t half as good (again, canon Remus : it makes sense from his flawed pov?)
Adding an important note:
When Sirius arrives at Godric’s Hollow and finds his best friend/brother and sister-in-law dead, his first thought was that he wanted to look after Harry, and because that’s what Lily and James wanted:
After realising Peter Pettigrew's betrayal, Sirius Black came to the cottage. He offered to take Harry, but Hagrid refused on Dumbledore's orders. Sirius then lent Hagrid his motorbike and went off to seek revenge on Peter.[Harry Potter Wiki]
So no, his first instinct wasn’t to rush off to murder Peter, his first instinct was to look after Harry. It is only when Hagrid refuses, and he knows he’ll only be able to take Harry by duelling him (which is too dangerous anyway, as Hagrid is holding the baby) that he goes after Peter.
(His second instinct is also not murdering Peter - he rushes off to confront Peter - he wants answers, desperately, to know why he betrayed his friends! Peter puts on an act, sobbing loudly about his tragically murdered friends and
“Then before I could curse him, he blew apart the street with the wand behind his back, killed everyone within twenty feet of himself — and sped down into the sewer with the other rats . . .”
He doesn’t even say WHAT curse he was going to use. Anyway, point being he was still in complete shock and didn’t really think through what he was doing)
TL; DR : basically every time he’s given a choice, every decision he makes, Sirius puts Harry first
😭😭😭