There is sooo much more damage occurring here than you think. Dwarves are first and foremost family first. Thorin making Kili stay back, you can see how DEVASTATED he is! "What did I do, why are you punishing me?" Fili sees this, which is why he looks pissed and defies his unlce. Kili is so crushed by this rejection he cant fight the infection any more and collapses.