On conquest and force but Robert was the only who's grandmother was a targ (and also seemingly the others didn't want it) Ned said as much in the first season when Robert was like you should have taken the throne and bed replied you had the better claim

If Ned had taken the throne he would still be king and there would be peace. No Cersei as Queen and her incest bastards, no lil finger at court or on the small council, I doubt Ned would have even kept varys. Basically it'd be no story cept the greyjoys rebelling