That would be pointless, because then just like PotD, people would just spam whatever gets them the EXP the fastest, regardless if they have the skill.
What is happening is that people simply do-not-want to play tank or healer because they aren't DPS, and they don't like being shoe-horned into a role that doesn't let them kill things quickly. PotD is outright mockery of the gameplay, because now you see people with straight 60's or 70's, and all they did was spam PotD, not play the dungeon content, and the reason they didn't play the dungeon content? Queues, and people being a toxic to each other about how to play the game. People aren't cooperating in PotD they are just running into rooms and letting mobs spawn all around them and if everyone dies, reset progress and start over, at most 15 minutes are lost. Faster than losing 30 from getting a bad team in dungeon content.
What does that say? That says that PotD is bad. It may be fun to do a few times, but it's an absolutely slog to use as a way to grind exp or tomes. SE made a mistake in having this content hand out rewards like candy. It should never have handed out experience. Likewise PvP should never have handed out experience. Now that both hand out more experience than doing the actual content, THAT is why there are queues. It may provide an alternate for DPS to level while there are queues, but what is the point if people don't learn the game mechanics?
What I'd almost suggest at this point is SE release a new healer and tank class and make sure it can NOT be leveled from PotD. Otherwise it will do nothing to solve content queues.
As for what, the answer for tanking is staring us in the face, create a Magitek "tank" job in the same idea that we have with the Magitek mount, only a bit more like D.Va in Overwatch where the player switches between gun-blade (DPS mode) and Tank mode (tank armor mode.) The plot line for this could even be as simple as Cid's Ironworks team creating a Magitek Knight (Tank) job, and the job storyline would involve some solo-instances into Garlean buildings to get the blueprints for thigns.
For Healer, let's look at a non-magic based healer for completeness. We have a one tank that uses MP, we can have one healer that doesn't use MP.