Well looking at the New Princess, there's tons of insecurities there.. With Kairi, her insecurities could be that she feels too weak to help and responsible for all that's happened to her friends, obviously she wasn't to blame, but she still might have blamed herself.
I thought the same thing. When Elsa came around, I kept thinking, "how on earth is someone that flawed a princess of heart?" I can understand Anna, who has similarities to Alice's willfulness, but Elsa? Characters with flaws are interesting, it doesn't make them one-dimensional, but the in-game mythos about the princesses just set them up to be absolutely perfect and flawless. Pure, charming, beautiful, willful, confident, etc. Elsa falls short of the mark. She's a princes by blood, but by heart? I'm still confused, but if canon says so, I'll accept it.
Rapunzel and her fear of defying her adoptive mother, but that doesn't really count, I guess, since tons of Disney princesses have done things to give their parents heart attacks (sneaking out at night for a carpet joyride with a boy from the streets, living with seven vertically-challenged but still fully grown men without blood relations, breaking curfew, meeting boys in the woods without asking permission, etc...).