First of all, King Jaehaerys sent two KG to the wall who'd abandoned his uncle King Maegor to join his camp. Other members were also sent to the wall or were executed because it was suspected that they had either killed King Maegor or let him die. So just because a KG abandons his King to join the rival King doesn't make it alright for them. And Barristan never stood up for what he believed in, he stood up for honour. Honour demanded he remain loyal. He didn't choose exile, he was driven out by Joffrey. If Joffrey hadn't done that, Selmy would have remained in his service by his own testimony.

The King swears no vows, owes no allegiance. He's King by divine right, by grace of the seven gods and that is all there is. It's not a constitutional monarchy.

As for Jaime, Just being gifted with arms doesn't make a Kingsguard. Many sellswords have excellent skills, you'd need to be as mad as Cersei to name one of them to the KG. You want men of honour to guard the King, whom you trust will keep to their vows and defend the King with all their might. Otherwise what is the point? A dishonourable KG would flee at first sight of trouble or join the enemy or kill the King. And Robert didn't owe any money to Tywin when he pardoned Jaime. The debts started after he emptied Targaryen treasury.

And no Kingslaying is a crime for everyone. When Aegon II's men murdered him before his defeat, his enemy Cregan Stark declared that killing an unjust king in lawful battle is one thing but foul murder is an affront against the gods who had anointed him. He had all of them including the accused KG arrested, tried and sentenced. Similarly, Eddard Stark wanted Jaime executed or sent to the Watch for Kingslaying even though he was Aerys II's enemy. There was no question of right, he was the rightful King by birthright, by divine right.

For Mandon Moore, Cleganes are the wrong parallel. Cleganes are sworn swords, not KG. As Tywin Lannister said of appointment of Sandor to KG "You feed your dog but you don't seat him with you at the table". You need your dogs to do the dirty work but you need lions to stand by the King. Once again as Tywin said, "Some tasks are fit for lions, others for goats and dogs." Lastly, if your rule is shaky, you need men of absolute loyalty and capability around you. People of fickle character will never stick by your side when the things turn sour.

None of your five (seriously?) comments are considering Robert's point of view, which is what this entire answer is based on. But to go through it anyway: (1) Doesn't mean Robert must refuse Jaime/Selmy. (2) Jaime provably showed to not follow regnal power above morality. Selmy is similarly honorable. (3) Jaime would logically be dedicated to defending Cersei, his loyalty is hard to question. (4) Jaime killed a king who Robert already opposed. Kingslaying is a crime but Jaime helped Robert. (5) Does not counter the deterrence argument I made. Also does not consider Robert's POV.

@Aegon: The question is "why did Robert pick them", not "did Robert conform to historical kings' approaches to appointing KG". The answer lists reasonable justifications for Robert choosing the people he chose.

@TheLethalCarrot: That has been edited out because I was indeed wrong about that.

Well if there are so many wrong assumptions and claims in one answer, it sure takes a number of comments to address them all.

Pity you don't get the option to move it to chat right away

Anyways you seem to have disregarded what I said. Not a single reason you mentioned is justifiable, for reasons listed already

Your first reason is that Robert had no reason to say no to men offering him their swords

But he had very good reasons to refuse their swords

Jaime's motives for what he did were not known to anyone except Jaime. There was no redemption, no question of sympathy

You want a KG who follows "Regnal power" not his own sense of morality, which would be good for men like Mandon Moore, as you say who'd be seen as obedient by Joff.

Selmy's case was slightly different from Jaime. We are in agreement over him that he is different from Jaime in the sense that at least he fought till the end

Jaime Killing Aerys didn't help Robert. Robert would have killed Aerys anyways. Eddard Stark was within the city moments after that happened. Jaime only stained Robert's throne

As for his loyalty to the Queen, loyalty to the Queen is not loyalty to the King. It's the KG, not QG