Personally, I don't have a problem with the current system where tanks and healers try to dps as much as possible. It suggests that healing throughput and tank cooldowns need to be nerfed, tanks need to take more sustained damage while main tanking in order to put more pressure on using tank stance rather than a cooldown in specific situations and most of all, DPSing on healers and tanks needs to have more of a cost. The problem is the barrier to dpsing as much as possible is quite low, especially given their current scripted fight design. But I don't think nerfing aggro would solve the problem either. We don't want either extreme. A patch where tanks have to sit in tank stance and only use aggro combo and healers only spam cures is not interesting, nor is a situation where healers are spamming Broil as much as possible. We want a middle ground, and one that encourages interesting dynamics between the Healers and Tanks in your group.
However, tank damage is too high currently, and to start addressing that, tanks need their offensive stances nerfed. Since tank GCDs are almost always going to be damage GCDs, they should have lower throughput in comparison to healers by default in order to encourage using Tank stance as a way to free up healer GCDs for damage, which I would agree is all the more reason to buff healer damage, but toning down Tank DPS modifiers a bit will help bridge that gap. That also means using tank stance should be less of a penalty, but it should retain its defensive boost, to make using that better in the long run.
Tanks should still be rewarded for stance dancing too. Of course, that starts with Grit and both PLD Oaths becoming oGCDs like WAR. That doesn't mean that DRK has to change, just that their damage bonus has to remain less pronounced with both Darkside and Grit up.
For PLD, I think reducing the penalty on Shield Oath and only making it apply to physical damage is sufficient, conversely rewarding them for sitting in tank stance during their Req windows. I'd like to see Sword Oath get a change where it actually boosts physical damage by a minor amount while the bonus auto attack potency is toned down massively.
For WAR, Deliverance is actually fine. What they need is some lifesteal while in Defiance to act as their own self-sustain buffer. This can be made weaker for AoE skills specifically as well. DRK doesn't need lifesteal, they have built in damage reduction. WAR has no passive mitigation without healer input currently, and passive sustain would fulfill a similar purpose, at least for auto attacks.
Furthermore, all tank cooldowns that explicitly reduce damage should be nerfed in potency, and lock you into tank stance for their duration (Note that this would not affect Shieldtron/TBN/Convalescence or any of the parry cooldowns). To compensate for this, they now stack additively with tank stance. For WAR this is a big mitigation nerf but that's kind of warranted at the moment. As another way to compensate for this, the lockout period will be removed if any attack they mitigated would have been lethal otherwise, and in the case of any personal cooldown, should reward an appropriate resource to the tank, that can then be translated into more damage later. IB refunding some of its cost would be an example of this. For something like Rampart, it could simply generate Oath/Blood/Beast Gauge proportional to the amount of damage dealt to you at the end of its duration, based on a percentage of your health.
With Shirk in the game, the off tank can translate the extra aggro they generate via stance to the main tank if they want to use a defensive cooldown, but it's unlikely that would matter. Instead, I'd argue that new abilities should be added next expansion to all tanks that share cooldowns with these skills, in order to encourage the off tank to use their time to DPS while they're free to, or in the case of PLD, perhaps make it easier for the main tank to tank in DPS stance.
In terms of theming, WAR is the aggressive/lifesteal berserker, PLD is the protective/supportive holy knight, DRK is the dirty/shadowy punisher. That does mean PLD/DRK should be better as straight walls than WAR, but they both go about it in different ways. That means reworking Shake it Off on WAR and giving them more ways to replace some of their cooldowns with more offensive utility (such as adding the old Berserk back in and making Inner Release a raid DPS buff). Give DRK a slew of personal debuffs to inflict regardless of stance, including their own offensive raid debuff and Slashing. Add more spells to PLD's utility belt, none of which need to include an offensive buff other than maybe Slashing, but could include another holy-themed DPS skill or two and maybe a change to Intervention allow the other tank to steal your cooldowns & stance for their full duration/effect, while fuelling your gauge in the process.