The shades dismounted from their ghostly horses. When they drew their longswords, it made not a sound. "He was going to burn the city," Jaime said. "To leave Robert only ashes."
"He was your king," said Darry.
It's paragraphs like this that make one easily forget Jaime Lannister's own sins. He is a highly complex character, after all, and what he experienced in his teens and early adolescence is enough to scar any person. I'm constantly perplexed how people can read these books and still swallow the propaganda surrounding Aerys' kingsguard. Oh, I have no doubt they were really great knights, great in fighting prowess and knightly honor. But they were rotten to the core.
To hammer that point home, let me make a rather strong statement: Aerys' kingsguard would have made great guards in concentration camps. They would have done their duty, not judging, because they made a vow. Their peers would hold them in high esteem, and if one of them had shot Hitler right before Auschwitz went into operation, they would be whitewashed and the assassin villified. It's understandable how conflicted Jaime feels about them, giving how large they loom as legends and that they mentored him. But again, the kindsguard is a rotten institution, and Jaime doesn't recognize the source of the rot, so we need to help him out.
Jaime is not wrong when he remarks to Catelyn that knights swear too many contradictory oaths. However, he never understood how to prioritize them. He should have listened to Brienne; the ghost of Ser Duncan the Tall always rides by her side.
The oath to your king is the least of all the vows you take, sers, and the oath to protect those who cannot protect themselves is your highest. You just need to look in a mirror to see why. You see yourself clad in steel, the best gold can buy, with deadly weapons in the hands of a body trained for one purpose, and one purpose only: to kill (Sandor had the right of that one). This is a privilege and a duty. A privilege because it takes scores of peasants working just to finance your training, your sustenance and your gear. A duty, because you need to earn that. And you can't ever earn this privilege by turning against those who afforded it on you in the first place.
This isn't rocket science. So why do knights in ASOIAF consistently fail to prioritize their vows right (as do the lords, by the way, but that's a topic for another day)? Simple: because it's easy. The kingsguard chooses to honor their least vow the most because it is the simplest and it comes with the smallest personal sacrifice. If you doubt this, read Brienne's AFFC chapters again ("No chance, and no choice"). But the kingsguard - and Jaime - fail this test repeatedly. And this is why you could use them to guard a concentration camp as well. It would be so easy, and you could always tell yourself you're just doing your duty. Tens of thousands actually did so before, and tens of thousands will do so again, because they fail this test, and no one will blame them for it.