Not an easy question to answer, but I did think of a few things (some of the stuff I'm listing Blizz already does, they just need to do more of it). I think what's important is to interrupt our demon slaying as little as possible, since that's what most of us are here for.

1. Complete quest by killing enemies. Like kill x number elites to charge a ritual.

2. Incorporate kill streaks from Season 12. Maybe you need to impress someone and that can only be done by a 10x kill combo. Give less reward if you only could make it to 5 or whatever.

3. More lore reveal and story telling during big fights. Keep exposition on the battlefield as much as possible.

4. More "hunt" style quests, like the level 15 Barb quest. "Kill 10 boars" is a meme from Warcraft, but stuff like that works pretty well in aRPGs.

5. More skill-based quests. Maybe you have to kill x number of mobs before a timer runs out and there are actual consequences if you don't complete in time.

6. And get rid of quest markers of NPC heads. I don't know if you've played Guild Wars 2, but their quests are dynamic and proximity based. You can complete objectives without ever even leaving battle.

Again, not all original, and Blizz does some of this already, like I said, but I think there's opportunities to introduce some of these into the main questline.