It goes both ways.

I love to micro-manage healing during a FATE, but I've found every time I start doing it I've got aggro coming at me from all sides. And absolutely no DPS or Tanks stepping forward to assist. At end tier FATEs, it quickly goes from healing other players to keeping myself standing and kiting as I quickly draw enmity from multiple mobs by healing multiple melee players.

I have no issues with the mechanics of that; healing draws aggro and that's fine. I do have issues with "Healers aren't doing their jobs!". Well, we can't heal when we're dead and the melee we're trying to help rarely get our backs in a FATE.