If we nerfed SCH's DPS rotation to practically mirror WHM/AST

[LegoTechnic]

And functionally infinite mana. And the ability to trivialize raid mechanics with absurd gigantic deployed shields. And multiple stackable percentage mitigation abilities. And cheating several extra solid utility abilities into their kit above the standard amount other healers got that required zero GCDs or cast animations from the character because Lily. And a constantly running regen that's more potent than regen before factoring Rouse in. And the strongest emergency burst heals if Aetherflow stacks permitted. And more mobility with Bio and Ruin 2.

Scholar was broken for a lot of reasons; the idea that the class had this one tiny sliver of an advantage in Cleric Stance is a meme. They need some buffs now though.

Fairy abilities all have a 2-second cast with accompanying animation that turns off the "regen" of embrace for a minimum of 3 seconds per use, and while the SCH themselves are not hampered by the cast times (outside of Rouse, which triggers a GCD), the fairy is riddled with its own responsiveness and range issues, as well as reduced potency due to it being a pet. Furthermore, the "regen" as you call it does not in fact activate if targets are over 80% health, and takes 2-seconds to cast when they drop under with a 3-second cooldown, and while it effects a single target it is difficult to control who it heals as it defaults to whoever has the lowest health in range of it (often not the tank in AoE situations). I understand you desperately want to make the fairy out to be some amazing advantage to justify your point, but it's not; Eos is three healing abilities that each last 20 seconds with either a 60 or 120 second cooldown (WD being 60, FI and FC 120) with a passive regen-variant that can be launched from a separate "platform" rather than centered on the SCH itself, and that's all it is. It's a neat utility that requires both placement management and a resummoning mana cost, but not something that somehow trivializes WHM or AST healing toolkits. In any situation where the fairy itself was "OP", a WHM could just as easily handled it with Divine Seal + Regen and Medica II, or in dungeons a Holy Stun, the difference being they had to be out of Cleric Stance to cast their heals. Prior to HW this was exacerbated by Lustrate being a a percent heal and castable while in Clerics, but this was changed at the start of 3.0.

Your other points were certainly advantages of the class, though. Mitigation was their role, and ogcd burst healing the compensation for not having any effective spammable heal (which they still lack). And while they didn't have infinite mana in raid settings (they still needed mage's ballad if healing and let no one tell you differently), they could literally DPS for days with constant low-mana cost casting which is something neither WHM nor AST were able to keep up for anywhere near as long without going out of mana. Most of that was due to the DoT ability focus, but the aetherize mechanic and energy drain were an obvious supplement for it. Regardless with their limited healing output and focus on burst heals gated by cooldowns, they were by design a perfect off-healer class, obviously able to support the primary healer's more efficient healing output with supplements and weakening effects. They made an excellent second half of a team until the party's gear level reached the point where the greater healing levels weren't required, in which case they could take on that role as well. And while HW sought to enhance SCH's ability to main-heal with gated cooldown abilities, their "bad things are happening" recovery still leaves much to be desired.

Technically all content could be cleared with two of any healer, be it double WHM or doulble AST or double SCH, but WHM/AST + SCH covered bases better and allowed for easier progression. You wanted the SCH because mitigating raid damage allowed you to get past mechanics at lower gear levels, and you wanted a primary healer for better HP recovery in between constant and repeated AoE booms. It worked out well, and if there was any actual problem in the design, it was that AST played so much like a WHM that it couldn't replace SCH's role in this meta as easily as it could WHM.