Arguably, Kylo wants the same thing that Vader always wanted. Vader wanted to rule the galaxy as family (father and son), and end the Sith with Palpatine. Vader absolutely hated Sidious.
They don’t destroy his mask, either, just the pedestal it’s on unintentionally. I’d say the fact that he’s keeping Vader’s mask prominently in his quarters as well as reforging his mask from TFA indicates that he does still want to finish what Vader started. Many people just assume that what he states in TLJ was him dismissing Vader’s legacy, but ironically what he is describing in that scene is EXACTLY what Vader sought, after killing his master no less (what Vader always wanted). He wanted to rule the galaxy with his wife and children, Sith and Jedi be damned. Trust me, watch Anakin talking to Padme on Mustafar and then this scene from TLJ. And we get to confirm this correlation in this film, especially if we replaced “and create a new order, together” with “and finish what Vader started, together” as the final line.