People think Sirius was brave enough to stand against his family's ideals. But I think it's that he wanted to be brave enough to stand against the "pureblood" beliefs. His courage was much more likely a response to his experiences, not some innate gryffindor bravery and recklessness.
It's like him being a rebel as a response to his abuse, him breaking the rules of the society he was brought up in, him being a troublemaker instead of a good heir like his parents wanted, it's more the want to be not like them, not so much as inherent bravery. Not to say that's not bravery, that is not the implication here. But people like to either diminish him as this glorified "good" character or make him utterly rash and irresponsible, and make that his entire personality, both of which are entirely opposite to what he is. Many just hate him as this "dark" character with deeds not worthy enough for redemption. Even in canon he was unjustly accused of being a criminal, and treated unfairly even after he was proven not to be one, by characters such as Molly.
The fact is he was reckless, and sometimes his actions did come off as careless. His actions sometimes make us uncomfortable. But his flaws shadow his complex personality to such a degree that people forget he's this brilliant, smart person who also suffered great injustice and is not just a "nice" guy, instead probably has this imo morally grey outlook on life, due to his experiences. His recklessness is a response mechanism, which might hurt others at times and I'm not excusing his actions either. The man spent 12 years in azkaban yet he was still sane enough to break out from there, which is a seemingly impossible task, the first person to do so. When he got the slightest hint that Harry might be in danger along with his burning desire for vengeance, he does the impossible, evidence of his brilliance and sheer determination and will, but also that hint of recklessness because he probably doesn't care about his life and later on doesn't consider the implications of his actions when he runs headfast into danger to protect Harry. He doesn't stop to think how that would affect Harry because Harry's safety takes priority for him, his utter lack of self worth easing him into the process. But there are moments where he's just this man trying to be a godfather to his godson, trying to be brave enough to be there for Harry, brave enough to try to live for him. Whether or not he wanted it to end at one point is completely up to interpretation, but I feel he tried to be there for Harry, wanted to be brave even when suffocating in the home he did everything to leave behind, his remaining memories that haunting him every step and reminding him of everything he lost. In a position like that, it'd be very easy to fall into despair and just lose hope, I wouldn't fault him if he did. He's far from being perfect, and that's what makes him so human, all his trauma, his flaws, his bad decisions, his hurtful actions. His story is one of the most tragic in canon hp, but canon reduces him to this reckless hot tempered side character, the black sheep of the Black family yet unable to delve into what truly separates him from them.
This also brings the point that Sirius "wanted" to not be like his family and by extension a Slytherin, similar to how Harry didn't "want" to be a Slytherin, tho Harry had at least canonically a more black and white outlook for life, but justified at the time, because at the mere age of 11, he learnt the evil wizard who killed his parents, the reason he was at the dursleys, the reason why he never had a good life, and probably never will, was in Slytherin, along with the dark arts and pureblood stigma associated with that house. Even though they all are just children, the encounter with Malfoy further solidified that anti-slytherin ideal. Although canon was crazy to make a single house the collective evil when it really is so much more (no one ever accused the TERF of being a good writer).
The way these characters (Sirius and Harry) are raised are also very similar. So gryffindor seems to value the want to be brave more than often, which might make sense as a trait actually favoured in a member of the house (as we see in Neville too), thus making some of these members kinda reckless to be perceived as more brave. Because that's what it is, the want to be brave.