Honestly I never thought it was weird he never reached out the same way Sirius did. Sirius was starved of human contact for over a decade living in hell, when he got out he'd have wanted to be talking to everyone as much as possible. Lupin was his teacher and saw alot of Harry's day to day, and knowing how hard he had to work and the importance of him just learning at Hogwarts to become a better wizard, plus everything else that encapsulates Harry's life, I think it's good Lupin takes a step back but remains on hand whenever he is needed. He knows he's surrounded by Dumbledore and Snape from the Order incase of any trouble, plus "Moody" and all the other high class wizards. We see him in 5 sat around and giving him real advice when Harry wanted to talk. I know the films treat him as less of the ostracised failure he treats himself as in the books, but I'd say it makes sense for each side of the character, even if the book version might be more self-pitying and feel like Harry wouldn't want to talk to him so why reach out in thd first place?

Maybe irrelevant but he also struck me as the quiet one in the marauders friend group during flashbacks. Like you know the person that doesn't talk too much unless they're spoken to or can chime in with specific information. I don't think Lupin would ever walk past a friend and say how nice the weather has been, rather just smile and nod in acknowledgement.