I think for healers it wouldn't be so much just healing but healing based around what their class is like.
Instead of just telling a AST to dps while healing. You tell them to maintain buffs effectively on a group of allies while healing them. That is a more worthy test of the player's use of the class.
Whereas for a Scholar, yeah tell them to effectively dps while healing.
White Mage I'm not sure.
Could do some of the tricks the WoW Proving Grounds did.
For DPS, they had multiple targets that needed to be taken out in a specific amount of time, including some that did not come in at the start of the wave. No damage to avoid, but had (I think) some mob AoE that would lower your dps if hit so meeting the enrage got near impossible if one didn't dodge. Promoted target switching and meeting a DPS check
For healers: they put you in a pt with other NPCs and expected you to keep everyone alive (and everyone could take damage, not just the tank. lots of AoE). If any of the NPcs died, game over.
For tanks, you had to pop your damage mit CD at the right time for big hits/mob waves and keep threat on multiple mobs at once. I believe you got a pt similar to healers but never did that one.
Had Bronze < Silver < Gold < Endless modes. Silver was required for Heroic dungeons (they dumbed down Silver a lot between MoP and WoD tho), Getting Wave 30 in Endless (was fairly hard) got you The Proven Tank/Assailant/Healer title.Your gear was ilvl capped too so you couldn't outgear it.
It wouldn't work 1:1 here due to class design here vs there (as an example most WoW healers have spell interrupts to mitigate some of that AoE, not the case here) but I think there's room for improvement in teaching people how to play their class. I think things like Savage Faust are pretty good checks for Alex but I'd like people to get a clue beforehand. Class quests alone aren't enough it seems, and when they did stuff like OG Steps of Faith and current Bismarck people would just drop the queue over and over again rather than deal with it (and even then most people who were bad and cleared it didn't learn, just got carried)
I wouldn't mind seeing some class specific quests like this for the new Relic quest line in 3.1 or w/e. Most of that relic stuff in ARR was busywork rather than actual "legendary" achievements, and since those are class specific I can't think of a better time to get people started.
In your example, targeting specific NPCs with specific buffs wouldn't be a bad idea for AST, to teach card effectiveness. WHM actually does a lot of DPS while healing, but they could also get asked to CC mobs with sleep or stun cause they have good options like Holy and Repose. SCH could be asked to move their fairy pet to specific areas to make them more effective or avoid cleaves (a big problem I saw with a lot of SCHs is being bad at pet maintenance). There's a lot of stuff they could do, and tbh asking a healer to DPS or keep an NPC alive isn't new for their class quests.