It's a combination of things. As a healer main, I can tell you that tanks complaining about bad healers isn't entirely unfounded. There is a significant lack of good healers in this game. However, there is a level of blame on everyone's shoulders. The tank needs to be sure to avoid all enemy AoEs, since in huge pulls, the healer often can't handle any extra incoming damage. The tanks need to be using their defensive cooldowns effectively. Meanwhile, the DPS need to be doing effective DPS, not just to clear the entire group quickly, but also to mitigate major damage sources first. For example, even if the DPS are AoEing the entire group, they should be focusing any single target attacks or AoEs that get weaker for each target hit on primary damage threats. In Bardam's Mettle, for example, there's a large bird mob on the way to the second boss that inflicts an extremely painful Windburn DoT. If it's a large pull, the healer often cannot take the time to use Esuna, and the debuff just gets reapplied anyways. So that mob needs to be the focus. To facilitate this, the tank should try his or her best to position that mob centrally so that it becomes a good central AoE target.

Also, I think tanks sometimes forget that healers have to dodge AoEs as well. A good healer is prepared for this possibility and can use its oGCD casts to keep the tank healthy while avoiding the AoE. Alternatively, they can position accordingly to minimize the amount they have to move to avoid the AoE, allowing them to use the brief time at the end of casts to avoid AoEs without significantly interrupting their casts.