I disagree. Allow me to explain it like this: the (new) purpose of Luke’s pep talk is to demonstrate that fear/hatred has impacted the decisions of him and Leia in negative ways, but when they were finally able to rise above it, that’s when wonderful things begin to happen.

Luke feared failure and the role of the Jedi in the galaxy in causing it. By the end of TLJ, he announces himself as a Jedi once more and inspires the entire galaxy. In this new chain of events, Leia feared the darkness that came from her father, and chose to forsake her Jedi path. However, when she sees Rey, a young girl with the darkness of Palpatine in her, she rectifies her past mistake. Instead of fearing Rey or hating her for what she can’t control, she takes up her Jedi journey once more and takes on Rey as an apprentice. Leia has learned from her own personal experience in life that she has the capacity to choose love over hatred; this isn’t something ingrained in her because of her father’s DNA.

This is what Luke is trying to tell Rey: follow Leia’s legacy of not fearing that you’ll give into hatred like your ancestors. You have the power to not fall into that trap, as some things are stronger than blood. In your post, you’re making the mistake of assuming that Leia isn’t a Jedi, but this just isn’t true. By training Rey, she has taken on the role of a Jedi master. So not only is she confronting the fear of becoming like Darth Vader again, but she is also confronting the hatred that she surely harbors for Palpatine. The fact that she is able to look past that Rey is a Palpatine and look at her as her own daughter is Leia’s legacy.

EDIT: Perhaps there is indeed a better way to convey all of this, but it would have to replace the line of “Despite losing everything, and everyone, Leia always chose compassion over hatred.” It would have to be something along the lines of “Leia confronted her fear by training you as a Jedi, and chose love over hatred”.