What I don't understand is, why the MC encourages ML to get revenge and at the same time w...

EXACTLY. I thought I was the only one who noticed, and she is being all happy olwoth the man she thinks is a murderer, no matter how nice he is if I am living with the person I think is a villain who killed an innocent child parents I wouldn't be all 'i love you daddy uwu'

I'm pretty sure it's meant to be exactly that- she understands the guy who would desire revenge, and with what's happened she could never tell him not to just because she cares for the person who committed the wrongful act. But at the same time, she loves her father and cares for her family deeply.
Her warring emotions stem from how she knows that while her father is the daddy she loves and cares for, he's also the demon ruler who's killed and wronged countless people.
I'm really looking forward to learning the truth behind the incident though, and really want to see what's gonna happen when she and everyone else learns that truth.

I might be wrong, but I think she means that she wants to make it seem like staying in the sect is the "better option" for him instead of escaping. Since if he truly wants to have his revenge, Wooryeon must get stronger first. Hence, "You have to work hard to build up your strength and have your revenge." She wasn't encouraging it, she's just following his line of thought.

Then, "I shouldn't encourage Ak Wooryeon from escaping the Demon Sect." -- because if he leaves, that would mean he would follow the original novel where he'll rally troops from outside to overthrow the Demon Sect.

After that she thought, "I will build up my strength to protect my family." She'll be able to keep tabs on Wooryeon if he continues to stay inside the Demon Sect, and she'll be able to prepare measures against other factors that would harm her family.

TL;DR Cheon wants to monitor and keep tabs on Wooryeon under the guise of "helping him" and making him stay inside the sect. This way, she'll be able to at least prevent the novel's plot from happening and protect her family.

You say that she want to keep tabs so she can protect her family, but what then, cuz he has a solid reason for wanting to kill her father, and she knew before hand what her father did but she took extra steps to deepen the connection, ultimately forgetting that she is living with two people who are destined to kill each other. I think I feel extra bad for the because he is being raised as the head of a clan he think killed his parents but finding out the child is not like the father, I am ultimately scared he will re think his revenge because of her because 'uwu I don't want to hurt her' it doesn't make any sense and would ultimately make his past trauma and life a plot point to just create a bit of drama when his concerns and fears are valid and SHE KNOWS this, but never thinks of this in front of her father, no person would be calm around their parents they think is an insane killer, I feel she still has this feeling of being in a book and not on a world where consequences, betrayal, and the like is real and can determine your future

I could understand this explanation but it doesn't seem like she is struggling with conflicting emotions. She isn't afraid or sad that her father is going to die in a few years. She seems happy to help him get stronger so he can omin the end I'll her father...seems strange. It's like we are missing a step in her reasoning.

I think so too, it feels like part of her thought process has been omitted

I'm very sorry if I am mistakenly reading your reply as aggressive. But I never said she's doing the "right" thing. I'm also not taking Wooryeon's trauma lightly, nor am I looking past the fact that the Demon Sect are indeed murderers. OP was simply asking why her intentions are contradictory in this chapter, so I was trying to share my speculations by trying to look through Cheon's (or the writer's) perspective. Which is why I started with the disclaimer: "I might be wrong".

"If she wants to keep tabs on Wooryeon, what then?", I don't know either. As I mentioned, she might be trying to at least prevent Wooryeon from escaping now that he's closer to the age where he started his revenge in the novel. Maybe that's why she helped Wooryeon literally escape when they were younger, because having him leave much much earlier might diverge the novel's storyline. We all know that didn't happen and he returned out of worry, so I think she's basically attempting the same thing now.

Personally speaking as a reader though, I think trying to make him and her father get along or "reconcile" is extremely idealistic, if not downright foolish. So I agree with you that she is currently looking down on the circumstances. I also agree that this might be another cliche where Wooryeon will stop thinking about revenge out of his feelings for Cheon, or something like that. Since this doesn't have the tags "drama", "mature", or "josei", I highly doubt the story would actually portray the dynamics between the characters in an objective or realistic way. But then again: I might be wrong, and if I am, then it would be a very welcome and pleasant surprise.

I hope this clears things up, I really wasn't trying to be argumentative or anything. I am also another reader trying to make sense of the story's direction. Cheers.

Indeed, in this chapter, the atmosphere was pretty casual and light-hearted when she was talking about helping Wooryeon become stronger. What I said was just another possible take on Cheon's intentions, and pretty much just giving the writer/author the benefit of the doubt. Hopefully we get more insight from future updates (⌒▽⌒)

Right, the chapter had a very cute and fluffy atmosphere so heavy thought didn't really fit it.
Your theory is also interesting and would make a bit sense. But at this point we are just filling in the gaps. Me too, hope future chapters explain her reasoning better(⌒▽⌒)

I think she just thinks him getting revenge is inevitable

I also wasn't trying to be aggressive or argumentative, it may have seemed that way since I am very blunt which gets me misunderstood alot, I am sorry, I was just confused with the manga is all, and as I was confused my words came out frustrated, and I also agree with some of your points as well, I was just genuinely curious as how the author would take the story since there are confusing plot point as of now

No worries!! Glad to know we were just both trying to figure out where the story is leading us ^^;;

Here's to hoping for some character development, because Cheon's logic has too many gaps right now, which doesn't make sense considering that she is being portrayed as "smart". I don't want to drop another promising series because of a questionable female lead again.

Saaaaame, i don't wanna drop another manga hahhaha.