There are a lot of people talking past each other in this thread due to a lack of experience with what "pushing a game towards story means." Most of that seems to come from a lack of experience with the kind of games that do this sort of thing, which leads to a question of "why would you even want to do this sort of thing?"
If you are looking to understand what's meant here, here are some games to check out:
- 13th Age. I recommend this one first, since it has a lot of D20 elements, and is also available through an SRD, so you can check it out for free.
- Blades in the Dark: a game centered on heists. If you think the "Dragonheist" adventure left something to be desired, well, this is a system to check out, since it's designed to do that sort of thing.
- Dungeon World: this is a PbtA game that's designed using D&D stats and terms. It has classes and levels and is really approachable, if you want to learn about story mechanics and keep familiar terms front and center.
- Swords of the Serpentine: this is a fantasy game based on GUMSHOE, a system about solving crimes and mysteries. Again, if you want to look at a system with mechanical support for that sort of thing, look here.
It's pretty clear that WotC is moving more in the "story" direction in the attempt to broaden the appeal of D&D and bring in players who aren't interested in the dungeon crawling aspects. There are a ton of people out there who might not be interested in going into a dungeon to get gold, but might find solving a mystery in a world of magic and dragons really interesting.
I think the idea of what can be done to make the D&D rules more focused in this direction comes from the fact that the system assumes you will interact with the skill or exploration system in a journey to get to the combat part of the game, and that combat and fighting things can solve big problems.
If you expect players to solve problems using their wits and skills without combat, D&D is (and this is my opinion only) a less than inspiring game, since that's not its focus. That's the point that at least I'm trying to get to: what makes for telling better story that doesn't lean on the combat part of the game.
So I'd suggest looking at something like 13th Age at least to see what these systems are trying to do. Maybe that will help, maybe you'll recoil in horror, but it's a direction that the designers really seem to be looking in.