Esuna is fast >_> I don't see why any healer shouldn't - only exception is if the status is about to fall off anyway.

When it's a whole-party status hit, I tend go - myself (unless I'm failing hard and the other healer is obviously much more helpful than I, and I acknowledge it) -> other healer -> Casters -> Tank/DPS

The only times I might not is if I pause to esuna, someone may die (such as if I'm trying to heal through heavy hits) or knowing the debuff is about to fall off by the time I finish casting (except I usually find myself hitting the button, anyway). Other times, if there's many people with multiple debuffs (such as WP HM, last boss) I'll esuna each enough to get rid of the more important debuffs before making the rounds to get the rest of the debuffs on all (such as the healing debuff because of Doom)