Personally, story-driven would be a nice change of pace because Mario as a whole is usually episodic in nature. Multiple games hardly ever create a narrative between each other, aside from small nods to each other sprinkled in. Even in actual franchises, like Mario & Luigi and Paper Mario, the titles don't affect each other very much, again, outside of small references here or there.
That, and episodic cartoons are the norm right now.
As for what I would like to see from a Mario cartoon... Personally, I'd like them to tackle something that has the same general tone or feel to the Mario & Luigi RPG series. Not that I think Paper Mario is bad at story-telling, don't get me wrong there, but those RPGs tend to focus more on side characters than the main brothers as far as characterization, and I'd love to see actual development from the two of them, something Dream Team tried to tackle (and Partners in Time made reference to). I don't need anything deep and gritty, and in fact I don't think that'd fit Mario in a canonical form, but lighthearted romps with some drama mixed in that pushes Mario and Luigi along and also explores their relationship sounds like the perfect fit for a cartoon.
Of course, the caveat to making the show more story-driven would mean that it'd have to have a set ending point planned for it right from the beginning, as opposed to an episodic form in which plots can just be thrown at the wall and see if they stick. I think episodic wouldn't be a bad route to take, it'd just be... Very safe. Story-driven would take more careful detail and thought, but as a result it would stand out more and it would be something the Mario series has never seriously dipped its toes into (as opposed to the episodic nature which has already been done in past cartoons).