Character don't need a sense of justice to be interesting
Sure they don't but they need something other than just being ****ed off and murderous 24/7. Otherwise they're just lame, one-note bores. Like OG Broly. That is why I went for the Hulk comparison (besides the fact that his power seemed to increase with his rage and that Hulk is just the more famous angry beast). The Hulk wouldn't be nearly as interesting without Bruce Banner, and the other way around. OG Broly is literally just a buff dude who wants to kill Goku and destroy everything in his path. That's what that character amounts to. Rage and power, full stop. He's not misunderstood as a character, he's just shallow as heck. As shallow as a character can be without being a straight up plot device in fact.
And yes you're right actually, Z Broly is indeed more similar to Doomsday, they're both just rampaging monsters with nothing really compelling about them, despite a backstory that could have made them interesting. Squandered potential.
That's why new Broly is a little bit more compelling than the old one. Rage is not all there is to him. He's got a "Bruce Banner side". Actual emotions that are not just hatred and anger. Emotions that led him to have friends, an actual support cast, and those things are actually what allow him to have way more potential than old Broly for character development. Now that Goku and him are on better terms, maybe they can spar sometimes (as Goku already mentioned wanting to do) and he can teach him to control his temper, but he's going to have a hard time with it, so they could play with the fact that he wants to be in control for the sake of his new friends but he can't. They could go with him accidentally seriously wounding Cheelai or the other dude because of this, causing all sorts of emotional turmoil in him, they could go with him holding a grudge towards Frieza, revenge stories are fun to watch. Perhaps Goku and Vegeta can intervene in all this, acting as conflicting moral compasses for him, with Vegeta being supportive of his revenge and Goku trying to make him desist, etc etc. Or maybe a story that doesn't involve Goku or Vegeta at all, they could explore the fact that Cheelai is now wanted and on the run, with Broly being her protector. Maybe a story revolving around teaching him the ways of a civilized man. Maybe a variant of this could stem from him becoming more trustful towards Goku after their frequent sparrings, which can evolve into the three of them moving to Earth, meeting Bulma and the others etc etc.
Because new Broly has a little bit of actual... character to him, that allows him to have friends and a support cast around him, which in turn provides him with way more room for development than old Broly could have ever dreamt of. Cause he's actually capable of interacting with other people in ways other than pounding them to a pulp. All of those ideas you had for OG Broly are good ones for sure, but they're not realistically possible or believable to be realized with how the dude was. For those ideas to be put in practice, Broly needs to possess some previously shown actual character qualities that can drive and justify such turns in behavior, otherwise it just comes out of nowhere and that doesn't work. But since the original is portrayed as this hell-bent on destruction killing machine and is given no other meaningful trait aside from that, there was no room for him to go anywhere as a character because he was always that one-note from the beginning. And even IF they actually meant to give him some actual character traits, then they failed miserably with portraying them which doesn't change the end result at the end of the day.