How can tanking and healing be made more broadly useful in more "role-specific" ways?

[Reynhart]

Who on earth ever told you that the fact you have HP means you need to kill stuff?!

All of our stats are designed for killing or not be killed, all of our skills are designed with for killing or not be killed. If you want winning without killing to be a major part, you'd have to change a lot of things in the game to adjust to this new goal. There is currently a gameplay where your skills are not used to kill, it's called "crafting".

And even then, I'm pretty sure that there are examples of NPC's not dying when their HP goes to zero. I can think of two off the top of my head simply from Kojin quests, then there's one or two in Amalja. There are even cases in story (for paladin, for example, where you fight that Elezen and that Roe at lvl60, but also in main story). Heck, Fordola is a recent major example...

Since you seem to like that word : Irrelevant. Even if we don't "kill" them lorewise, the goal is still to deplete their HP bar. Especially with Fordola, in case you didn't realize we were at war.
And if we remove that goal of "Depleting the HP bar", suddenly, DPS jobs lose their purpose.

The only reason why the game is all about killing is because it's MADE like that. There is zero reason, story-wise or anything, to not take advantage of that.

The gameplay is made like that, the story is made like that. To change all of this, you'd have to make real core modifications far above what could make tanking and healing more interesting.