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

[Shurrikhan]

I think it's not enough by itself. For me, the problem with stances is that the damage difference between tank stance and DPS stance is too low. If you increase damage regardless of stances, this difference would stay the same and once people are accustomed with the higher damage, they'll go back to DPS stance...unless the increased damage is so high that they can't do it anymore.

Fair enough, but let's consider: An increased penalty on "DPS stance" functions identically to reduced passive mitigation and a tank stance strengthened to its previous eHP levels. Moreover, no matter how great the gap is increased, there will undoubtedly be a point at which it's worth and a point at which it is not. At present, gear tends to favors avoiding tank stance, as it receives no benefits that DPS stance does not capitalize upon further (if tank stance eHP was sufficient at lower gear, then it will be excessive with additional gear), but at any given point as DPS gradually and increasingly outscales percentile mitigation (since encounter damage does not increase with your gear, and therefore neither can percentile mitigation) that decision is straightforward, and probably ought to be. Making one stance intentionally punishing won't change that; it just bloats the ability and introduces counter-intuitiveness.

Sure, a mandatory tank stance would indeed put more emphasis on tanking, but it would piss off the whole population that loves agressive tanking. For me the sweet spot would be that damage are high enough so that a healer paired with a DPS-stance tank would had to focus more heavily on healing, and the mitigation on tank stance would be high enough so that a healer who likes to DPS could do that a lot more when paired with a turtle tank.

So no types of tanks and healers would be left behind.

I've never recommended a mandatory tank stance. If it's mandatory, it's not a stance, but rather just bloat (in all but maybe solo play or in the destruction of defenseless objects). I'm just saying that the more something appears to be made effectively mandatory (optimal by a notable margin) not because of universally applicable scalars but through appended punishments, the worse it feels.

And note the situation your broaching is in every way a compromise. There's no point at which no one is "not left behind" unless you consider the most marginal presence or satisfaction as complete inclusion.