I specifically did not use the phrase "the core responsibility" because it is indeed one of a couple. But (holding aggro and directing damage to oneself) is as simple as activating an aura. Considering they just did a round of increasing tank aggro it doesn't seem they're interested in bringing this back as a meaningful mechanic, assuming it ever really was. And "to try to reduce (damage)" is included in "alleviating the burden on a healer", whether by mitigation or restoration. There's no real functional difference between TBN shielding and BW healing back incoming damage. Yet one is OK and the other is not according to your phrasing?
I've never considered off-healing as part of my role design as a tank. There are either support or survival abilities plain and simple. Could they be abused? Sure, lots of thing have been and still are, in this game and many others. So yes things need to be continually analyzed and rebalanced accordingly. But realistically there's probably never going to be a point where healers only heal and tanks only hold aggro. And to be honest there probably shouldn't be.