If Rey believes Palpatine will kill her friends, and Kylo Ren destroys her key to finding Palpatine and killing him, it could be enough to warrant her anger.
Basically you’re just changing “Palpatine killed your old family” to “Palpatine will kill your new family”, so if you feel like Rey doesn’t have enough motivation to attack Kylo in that moment, then it is just a problem with the theatrical film.
I’ve had some other ideas, like maybe Rey has vision that Kylo will be the one that will kill her friends, which is why she wants to kill Kylo, and maybe Kylo is trying to tell her that Palpatine is wanting tricking her, wanting to turn them against one another.
Or, maybe Rey’s vision is that she will kill her friends. Maybe in the first vision, she just sees the lightning, but doesn’t see Palpatine. But then during her Death Star vision, she realizes it was her who created the lightning (her lightning on Pasana could foreshadow this). Maybe in the vision she is on/next to the throne with Kylo, so she feels Kylo is part of the reason why she turns to the dark side in her vision. So if she kills Kylo, the vision won’t come true.
So, basically Rey has a vision that her friends will die by Palpatine. But Kylo also has a vision of Rey and Kylo Ren killing Palpatine and ruling together. But on the Death Star, we discover this is the same future, and it is actually Dark Rey that kills her friends in the final battle.
Just brainstorming right now.