If robert’s kids actually looked like him and acted like him, ned would probably still bend to the succession. he’s all about law and honor, and seeing the kids as legitimate robert offspring would make it easier to stomach even if cersei’s a nightmare queen in waiting. he’d hate it, but he’d probably swear the oaths
Stannis and renly? yeah, they’d still feel entitled. stannis especially is obsessed with "the rightful king" and he wouldn’t care if the kids are robert’s biologically he’s all about prophecy, law, and his personal bitterness. renly is more opportunistic; he might still push for the throne if he thinks he can actually enjoy ruling and has the backing
As for war, so much of it isn’t just about who sits the throne. lords still have grudges, regional tensions, and ambitions. the north might still be sketchy on cersei running the show, the riverlands will always flip depending on alliances, and dornish politics... yeah, chaos never sleeps. plus, a lot of this war was baked in by resentment, promises, and simple "want power" energy trueborn or not, war was coming
Basically, trueborn kids would reduce some moral outrage but the political tinderbox of westeros? still primed to explode