Just to put my two cents in.
To get what those involved in the healer strike is asking for is to gatekeep healing. The ego one must have to claim that "No role in the game pushes this skill more than healer", what a load of crock honestly. And I say that as a healer main who is quite skilled herself. It's like comparing apples and oranges. Tanks have to do a lot to ensure they stay alive, because guess what, in raid settings they are not going to survive on their own without us. Likewise, if they die, it is unlikely that we will survive because those auto attacks the tanks tank pack quite a wallop. Sure you get the occasional quick save with healer LB3 or you might have someone particularly skilled able to raise the tanks quick enough to survive such a thing, but it is not often. And of course the DPS need to master their rotations to be properly effective.
With regard to the dungeons, there is a reason why dungeons are so easy. It's for the casual crowd. Highly skilled anyone is going to be bored in those dungeons, period. They made the content more difficult this time around, but it's still "easy" enough so as not to be overwhelming to casual players.
Regarding the other roles having self sustain? Melee and phys ranged get what, second wind? That's barely a quarter of their HP, if that. Bloodbath is only available for melee, and casters? Ha, nope (red mage gets vercure but let's be real, that falls right within the lore). Also, I can't tell you how many times I've seen a red mage save the party when BOTH HEALERS DIE. As for Tanks, warrior's whole shtick is literally being "warrior smash, warrior self sufficient." Paladins have holy magicks after a fashion and frankly reminds me most of war cleric in D&D. Dark knights don't get much in self sustain—they get Abyssal Drain which is only really effective in AoE, and then living dead, which they get every 5 minutes and they have to be healed to full or else they die. I'm not so sure about gunbreaker and their capabilities, but i've never seen a GNB finish off a boss all on their own with no one else. Even when I do see a tank pull it off on their own, I cheer them on. You know why? Because they are succeeding where EVERYONE ELSE FAILED. Frankly, the fact that you have to watch a tank do that sounds more of a skill issue and less of a design flaw.
Also, I love that these classes have self sustain. It helps ME out as the healer, because I don't have to be like "oh s---" and trying to save them because they stepped in bad. I can effectively triage who needs what.