I personally have always felt that, on Jason's end, the rivalry he bad with Percy was done to like meet the expectation of his role S a son of Jupiter. He doesn't even know how to have a genuine rivalry because he's all in his head.
I dunno I might be talking outta my ass
i understand how it comes across that way. honestly, that's probably how rick wrote it. "well these two are kids of the big three, gotta put some tension there."
he clearly didn't think about percy and jason as three-dimensional characters that could have a rivalry grow naturally between them. he thought a rivalry would just be a flat plot point and made something up.
the rivalry should have been jason trying to fill percy's shoes at CHB for months, and still falling short because he's not thinking of percy as a person who can't be replaced rather than just an empty leadership role that needs to be filled.
the rivalry should have been jason learning that percy became praetor after spending less than a week total at CJ—he was literally on a quest 90% of the time. no matter how much he helped in the battle at the end of the book, jason wouldn't think he'd truly earned his position. earning a praetorship, jason would think, takes years of work, not a week.
the rivalry should have been percy remembering how bitter he really is towards the gods as he recovers his memory, percy realizing that CJ deadass has an entire child army, fully sanctioned by the adults in new rome, and realizing that those adults are just as bad as, if not worse than, gods who would send children on life-threatening, world-saving quests, and the only person he can really take that repressed anger and bitterness out on is jason.
the rivalry should have been jason hearing how flippant percy is when talking about these deities that they're meant to worship, how much he insults them and how impertinent he can be without consequence, and he simultaneously envies how much percy's able to get away with and hates how disrespectful he is.
we should have seen roman leader jason, groomed from his toddler years by the legion and lupa herself, who leads a structured, militaristic camp that prioritizes the safety of the many over the safety of the few, bashing heads with greek leader percy, who went from being an outcast to growing into leadership because he wanted to save as many people as he could.
we should have seen jason's envy of the fact that percy had a mother. we should have seen jason's jealousy over how much percy's father loves and values him. we should have seen jason's heartbreak that this self-important, standoffish, disrespectful jerk has the brotherly relationship with thalia that jason's desperate to have.
we should have seen so much more of these two characters that made them seem less like legendary heroes and more like the traumatized teenagers they are.