Coolasiangirl

The emperor favours that son that much because he is the son of the woman(late empress) he loves. The emperor hates ML's maternal family, so he hates the ML.

In short, he gets a happy ending.

If you want more spoilers on his fate, then here:
He did a 180 and became a "good father" to ML. Last we saw of him, he was chasing after/trying to win the affections of ML's mother, who wants to leave the palace to see the world. I guess that implies he will spend the rest of his life focused on his romance?

I know it is only in the beginning that ML wanted to kill her, but that is enough to kill any chance of romance between them. I'm speaking from a real-life perspective where attempted murder is something you can't come back from. And regardless of the ML's feelings, FL does not owe him a relationship. She told him repeatedly no, and he should respect that. Again, speaking from a real-life preceptive.
This is not real-life, so I don't really expect it to follow these morals, but any frustrations you feel should be directed at the writing, not the FL. The FL is just being a rational human with self-worth.

First, he did try to kill her in the beginning. Just for that FL is in her right to want to stay as far away as possible. Second, no one ever owes anyone a chance. No means no.