I'd give them more credit than that. A noble who went through the whole squire process with a good knight and didn't fail at it would get some fighter levels. If you participated in a war and survived, a noble cold get to... level 5 fighter. More if you fought a lot.
The weakness in them is their starting scores. Even for elities, their stats would suck. Few knights had Str over 14 and few characters had positive Charisma modifiers at all. It was their levels form squiring and armor that isolated most normal nobles from common rabble.
Which brings the issue to the PCs. In ASoIaF, personal Charisma doesn't help you in battle nor help you lead. Look at the Baratheons Brothers for proof. The smallfolk fight due to your family name, your personal/family wealth, or favorable odds. Nobles the same with more emphasis on the latters for some and the formers for others. So a Charisma based class just doesn't fit all that much except for a select few who can't be adventurers (Faith Militant and Kingsguard). In this world, they use strength, experience, toughness, and skill to fight. Charisma is for outside of combat.
I disagree. I will try to avoid real spoilers by being vague, assuming you've read the books. Speakng of the Baratheons, people fight for Renly at first because he has Charisma and Stannis doesn't. In order to correct this problem, Stannis has to use EXTREMELY extraordinary means. Personal Charisma is definitely a factor in the success of Jaime, the Knight of Flowers, Ser Barristan Selmy, Ser Arthur Dayne, etc.
Also, this class is no more Charisma-based than the paladin, on whom it is based. If I were making one, I would definitely want Strength to be my highest stat.
Maybe Conviction Dice is the wrong name. When I talk about the knight's conviction, it doesn't necessarily mean a code of honor or higher morale belief. It is just being convinced that you deserve to win and that you have the advantage over your opponents. You are more effective not just because you are formidable, but because your enemies believe you to be formidable-- this is why the knight's Smite version does psychic damage. You are a terrifying armored dreadnaught bearing down on opponents who mostly are not. When you fight another knight, it is a the stuff of stories, a clash of theoretical titans.
Each knight's Conviction is different. Early novels Jaime Lannister is driven to victory by loyalty to his family and absolute faith in his own superiority. Brienne of Tarth, more like a traditional paladin, believes in the rightness of her cause, and has the additional drive of proving herself in a world where women don't become knights. Jorah Mormont believes absolutely in his khaleesi, loving her but also thinking she offers something better for those she would rule. A Lannister knight in the later books is driven by his fanatical faith. Barristan Selmy and Loras Tyrell fight for honor, loyalty, and to embody the ideals of knighthood. Ser Gregor Clegane believes absolutely that might makes right, and that his extraordinary physical might will drive him to conquer anyone who opposes him. PC knights would be like these examples, not like run-of-the-mill knights. Conviction dice represent the qualities described above.
Obviously, I will need descriptive text describing all this.