ehh it's just me musing but. I do find it a little curious that (depending on who likes whom I guess) cora is usually either seen as some saintly flawless goofy figure or a brainwashed cop who got killed because he refused to try and save his brother. I do think his character is not exactly easy to pinpoint, considering he appears for a very short time and in a flashback nonetheless, plus the fact that he's dead means no further fleshing out of his character - broadly speaking - outside of the people who knew him and are willing to provide insight on what he was like, though that'd still be kinda biased.

however I believe there's actually a bunch of things that were straight up shown and some that can be pieced together from what little panel/screen time he had: ie how he's seemingly more bad tempered, impulsive and violent than he's portrayed as in fanon at times, albeit motivated by his own priorities at the moment (trying to kick the shit out of law to stop him from exposing cora to doflamingo) + his sense of what's right and wrong and to whom it applies (punching medical staff and setting hospitals on fire for mistreating law, whom he'd been trying to help).

specifically about the navy and doflamingo part... it irks me a little, tbh. partly because it removes what little agency cora had during the flashbacks and sort of waters down his motivation to stop his brother. it's not just whatever crimes doflamingo was committing or planning to back then and the navy wanting to put a stop to that - the thing is that cora was influenced, at least to an extent, to oppose doflamingo based on their childhood experiences with (ofc) the more negative ones, which include doffy murdering their father right in front of him, overshadowing anything else. as he tells law, cora can't fathom how their kind parents could've borne someone as evil as his brother. and yet. that's the other thing. cora was very much a child, and younger than doflamingo at that, when the elder DQs chose to leave marijoa and all that it entailed after. between all the traumatic events he lived through and later being raised by a marine (sengoku of all people), I'll be the first to say his perception of those events, of their parents and doffy himself is not really the most unbiased or reliable. we don't see him questioning the existence of celestial dragons (beyond warning law he's in danger when cora finds out about the D) nor the nature of the WG/the marines and the antagonistic role they play in OP's universe. we didn't have him long enough for those things to be put to question deeply anyway, especially not wrt to doflamingo, so imo it makes sense that his focus wasn't on "saving" but stopping him.

that said... he does witness the worst of it, kind of. through law. law is the very reason why I don't agree with the idea of cora being simply a brainwashed cop. this guy watched how people (those who should care) mistreated, dehumanized and demonized a sick child over prejudices caused by the lies the nobles and WG itself relied on to sweep their own corruption under the rug. he saw first hand how all those doctors ran to call the WG to kill the child and how they answered to do that. and what did he do? he lied and betrayed the organization he'd been part of (presumably for more than the years he spent undercover) and the man who'd raised him like a son just to save the kid that everyone, even the so called justice, had turned his back on and would've gotten rid of if given the chance. heck, when he first brought up the topic of law with sengoku, the man basically told him not to favor him too much for it could jeopardize his mission.

but perhaps the biggest proof is that he lied to law about being a marine when the latter directly asked if cora was one. as he later admits, cora lied to him about this because he didn't want law to hate him - and knowing all law lived through (flevance), seeing some of it himself (their hospital shenanigans) and what law told him as well, cora knew he had plenty of understandable and justified reasons to hate anyone ever slightly associated with the marines or the WG, including cora. to me, someone who's completely blinded by the navy/WG propaganda and follows their every order to the letter without thought wouldn't have denied his own affiliation nor been so determined to ditch being a marine and make an enemy out of those institutions (even if that also meant betraying his father figure) just to save, protect and do right by a child who'd been clearly failed by them. at no point did cora ever try to argue that Not All Marines, much less express any other sentiments of that sort to law.

on a similar vein, despite insisting doflamingo was evil and an agent of destruction - law is also the proof cora was somewhat aware that his brother (and people like doflamingo) normally don't pop out of nowhere and do Terrible Things just because. that maybe in other (better) circumstances, doffy might've become someone different and/or made different choices. after all, cora is the one who points out the similarities between doflamingo and law, and eventually does his best to turn law's life around so that he won't follow the same path. should he have tried to save doflamingo as well? when? how? would it have worked? who knows. and if you ask me, regardless of their similarities at that moment in time, doffy was already a grown ass man compared to law and cora himself was just an even younger kid when shit hit the fan in their childhood. I'm not sure doflamingo (as an adult) would've been particularly receptive of "help" either, considering his disdain for the kindness in cora and their father that he saw as a weakness. not to mention waaay too many other factors that come into play also (trebol and co's grooming and influence for example). still, one of them did pull the literal trigger in the end and it wasn't cora, so there's that.

all in all, for a character with such a short lived amount of time in the story - cora is quite the complex one and so very compelling. characterizing him as just strictly one thing or the other can be a little reductive but the fact that his character can be explored beyond that in the first place (once more, despite his lil bit of alive and onscreen moments) is what's fun and says a lot about the writing itself.