If possible through undermechanics (e.g. something as freely alterable as positioning during GCDs without positional modifiers), I'd prefer everyone have some responsibility in enmity management. But I would hate to see it returned as just yet another CD which 99% of the time we hit immediately upon its refresh. I'd rather avoid any {Invigorate, but for X} style skills altogether except where they enhance the feeling of rhythm and aesthetic of the class in use that is otherwise free of any consideration.
Personally, I'd also rather mitigation play a large part in the alternative to pure damage, rather than merely enmity. Of course, I find Enmity as it stands only a derivative mechanic, as there is no real sense to enmity except a modifier attached to damage. If Enmity actually had more of a sense of something that you can and must edge out as more than just a single point's lead and situationally some safety margin, then my interests would likely shift, but for now it feels like mitigation, not enmity, ought to be the opportunity cost of damage, given that, say, punching someone in the face tends to piss them off more than their guarding their self against the same.
As for the comparison of healer and tank maximal damage, I'd prefer both be nearer to that of DPS (with DPS in turn having a little more flexibility in their rotations if they ever play a larger role in mob manipulation or damage at a given point in time comes to have more impact, rather than just total damage dealt). In that reduction of the output gap, though, should also come a decreased gap in offensive complexity, and I honestly think healers could see their way clear to have a little more complexity in their offensive play without feeling significantly interrupted if their tools were just revised slightly for more flexibility and damage-interplay (without tight or specific timings). Tanks' and Healers' available active/choiceful throughput should reflect their costless throughputs (e.g the strength of passive mitigation and oGCDs by situation and total healing dealt or made unnecessary). If a Healer's oGCDs outperform the combination of a Tank's passive bonus mitigation, the situational advantage of their added eHP, and their mitigation oGCDs combined, then that tank should have the higher damage over time when each have 100% optimized damage uptime. The same comparison could be made between healers (to other healers) and between tanks (to other tanks). The idea should be that each brings an equal amount to the table, though the manner and base allotment of their outputs should differ. (The job with the least passive role-performance should have the most flexibility in what it does, which may itself be a minor advantage where it allows for more situations by which to survive, and the job with the most ability to respond to or counter situations of extreme risk would have the lowest total impact over a more sustained fight, if only slightly -- etc., etc.)