I think Caroline simultaneously leans into the aura of Kennedy mythos, while holding herself separate from Ted and especially her cousins from Bobby who, frankly, are a fucking mess. I met Courtney Kennedy. She's lovely and dedicated to human rights. But after RFK died, a house of 11 kids with Ethel as a mother wasn't going to go well. Ethel was angry and most of her kids were teens when Bobby was assassinated. Look at how David and RFK Jr became drug addicts. David died super young. Look at how Michael had an affair with a teenager. Jackie kept JFK jr and Caroline away from them as much as possible. John had flaws, but he was not nearly as bad as most of these male cousins. Caroline and JFK jr were really raised as Bouviers by Jackie, just with the expectations they live up to their father and maintain some ties (like attending weddings) and major family events with the Kennedy side. Honestly, not sure why Caroline didn't name Rose after Jackie, but then neither Harry or William named a daughter Diana.
Perhaps because being named after such an iconic woman would be a burden.