SCH's damage and healing as always been lower than WHM, and healing and MP management lower than AST. SCH got the position as off-healer/DPS Healer because it was practically designed for it, not because it was some overpowered monster.
SCH's DPS rotation was largely DoTs, which have low MP costs. That combined with part of it's MP management tools (Energy Drain) actually being part of the DPS rotation, meant that SCH could DPS indefinitely and be nearly MP-positive while doing so even if thier actual damage was relatively low, as opposed to WHM for example, who could output much higher damage, but only for about 2-3 minutes before they ran completely dry on MP. As for actual heals, SCH only has 3 on the GCD. Physick pre-SB was just a straight Cure clone, with the same potency and MP cost (it now still shares the MP cost afaik, but is 50 potency lower). Adlo and Succor both have absurd MP costs for how little they heal, and while they do have the shield aspect, that doesn't exactly help when trying to top people off. The only other heals SCH has are tied to Aetherflow, so are gated at 3 uses between them per minute, but that allowed them to quickly drop cleric and pop them between attacks. A SCH in HW who was spending significant time healing would go out of MP in about 1:30 on average, so I don't know where you got this idea that they had MP efficient healing.