Okay, I get the logic of a great deal of this, but I do want to play devil's advocate, because I find this conversation fascinating. No hard feelings? :D
Rek'Sai is a mature Void Creature, and many of the older creatures (i.e. Cho'Gath, Vel'Koz, Kha'Zixi) have motives to be in the league. Cho wants his influence to derail the world in order to provoke havoc, and so wants to win as many matches as possible so, I don't know, nations want more Void creatures to come into the world to be used as weapons of mass destruction. By participating in the League, Vel can destroy and analyze as many life forms as he wants, and he wants to know EVERYTHING. Kha wants the thrill of the hunt and to consume everything he can so he can hunt more, and thus finds purpose in the league by fighting Rengar. Perhaps Rek wants the League to be the arbitrator of her territorial claim in Shurima. This would also explain Azir's motive for being there, since he probably would have a tough time rebuilding his nation with a Rune War occurring in his country.
Isn't Aurelion Sol under the control of Targon? Couldn't they just order him to compete in the League? And Bard is a protector of balance, so preventing another world ending Rune War would be in his interest, right?
Kindred I couldn't explain. Game, set, match on that one.
Aatrox is less BLOOD on the battlefield than "glory, no matter the cost." If he wins every match on the Rift, he is the best fighter to ever exist, never mind the fact he probably just destabilized the political situation of an entire continent, leading to more of his favorite thing: conflict. What better reason to fight than that?
Syndra is probably one of the few champions who I've ever felt was underdeveloped in lore, so I couldn't say what she might feel.
The Harrowing argument I will also have to surrender to. But the current lore does not explain why Nautilus was seen destroying an entire cardinal direction's worth of Harrowing Ghouls, might I add, where the combined might of Gangplank and Miss Fortune only barely held off another (man, Nautilus is awesome). So the new version has holes in its logic as well. I mean, Nautilus shouldn't even exist in the new lore.
Sorry, that last paragraph was an irrelevant rant. The point is that you win on the Harrowing.
If anyone else would like to chime in on anything I missed, it would be greatly appreciated.