I see what you're trying to get at, but I don't think it would work. The problem is people think tanking and healing is boring, just taunt and stand still, just lay down all your healing things and maybe use a burst heal every 20 seconds or so. It's not engaging for people who don't normally do it.
Then for people that are healer and tank mains (like myself) it can get frustrating babysitting a bunch of people that go and stand in red/step in front of the boss and get oneshot and blame you for it.

Adding xp/achievement rewards aren't going to change anything just like no one is master fisherman even when they brought in perfect roe.
What needs to happen to encourage it more, in my opinion, is make it fun, engaging and rewarding. How? Make dungeons/trials have more engaging mechanics with satisfying rewards. So people will want a good tank and it will be fun tanking.
Which we already kinda have, each additional dungeon/trials added have had super fun mechanics added.
I guess tank mains will always be tanks.
Dps mains will always dps.

(But they also need to adjust solo questing for healers and tanks, that stuff sucks and takes so long, boring as hell)