My kids and I loved Ziggurat. It's a legacy adventure game about climbing a magical ziggurat. Each time you win the game you unlock significant changes to how the next game is played and progress the story of the game a bit more. There is a genuine feeling of exploration and adventure.
Some reviewers panned the game for being too simple. Yes, there is not a ton of strategic depth to be had here. But my kids play Dominion and Forbidden Desert and Innovation, and they had a blast with this game. It doesn't need to be a brain burner for them to have fun working together and climbing a cool cardboard pyramid. Also, the game packs in lots of thoughtful extras that it could have done without but make the experience even better.
I have no idea how available the game currently is. It might be hard to get.
We also play a lot of No Thank You, Evil! Like you, I do a lot of story-telling with their input and sometimes dice. I steal a lot of ideas from No Thank You, Evil! and sometimes use their character sheets and modules when we've got extra time.
Finally, I've had a good experience playing The Quiet Year with them. It's a collaborative map-drawing and story telling game. I curate the cards a bit, since some of my kids are young and some of the cards cover more mature topics. And I guide the early collaboration a lot to help them get past the world-building step and into the game with a setting that will spark their imagination. After we play a session, we have a cool map that I then drop into a frame and hang on the wall until the next session.