The reworked Kayle has more human origin rather then the Kayle from the entire race of angels. She was born in the time of great, lawless unrest, when people were killing each other like it's Mad Max.
Imagine spending your teenage years knowing you have to defend yourself and your family every day because people out there want to kill you (at best) and take your stuff and no king, no rules, nothing would protect you from this.
Then you get the power to be basically unstoppable, the power to change the anarchic society that preyed on its groups and your family. What would you do?
First Kayle defended her settlement, then she actually started gathering people and coming after the tribes who would have came after then. She was establishing and spreading the law and order. Also she was making sure that people follow the laws because she believed that the laws were the only thing capable of preserving a person. Given that Runic Wars were one of the darkest periods of Runeterra, I can imagine she was not nice about it because people don't just stop being assholes to each other when asked nicely. She also held herself to a very high standard as people were looking up to her.
Maybe back then Kayle thought that she is doing it for her family: she did let Morgana redeem criminals who were repentant and she never forced her to follow her example. Maybe she believed that while she is vigilant, Morgana can be free from the pressure of having to be a paragon.
In spite of all her harsh words, she never blames Morgana for what happened to her father and she doesn't even sound pleased if she kills her: she basically says that Morgana was too good for this world.
So, she still cares about her sister, but she will never agree with her about her mission because Kayle was the one bringing order to her settlement and beyond, therefore she probably saw a lot of disgusting and gory things (slavery, pillaging and apocalyptic bugaloo). This is why she believes that people are rotten and will live off the suffering of others if left to their own devices. Only if they are given a fear of judgement, will they look back on their ways and thrive towards a better society.
