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People are invested in Jaime Lannister and Darth Vader because they are human, their story, their motivations gives you an insight into them, people can understand why they became monsters.
Let's take the motivations of Jaime Lannister, grows up, has a distant and cold father, his mother dies in childbirth, is more connected to his twin due to it, engages in a twisted romantic relationship with her, knows it's wrong, desperately wants something good and pure in his life and joins the Kingsguard, discovers the he serves a homocidal tyrant and at a moment of crisis realizes that even this good and pure notion of duty is hollow because duty demands that he become an enabler of infanticide(all the children of Kings Landing).
To the audience he's human, his motivations are human.
As for Tybalt and Burke, they are not the main antagonists, they are obstacles at best, the interactions of the protagonist with them are limited so far to being mere hurdles in his path towards his goal, his goal is family, to belong somewhere, who makes that less likely, Tybalt? Burke? Quinn?