From the start, being the DPS healer was bad for the health of the game. You couldn't play high level content without a SCH if it was the "DPS Healer." So that bit needs to go. Fortunately you can get rid of it by simply equalizing each healer's ability to do damage, rather than nerfing SCH. FFXIV is designed so everyone HAS to contribute to dealing damage in high level content anyways, so nerfing this would be antithesis of the healer role that SE had, intentionally or not, created and sustained. Getting rid of this would be tantamount to removing the entire healer role and replacing it with a new role that simply has the same name. It's a quick way to lose an entire segment of the player base.

The other aspects are things we can do something about. Being the pet healer is hell for SE from a service standpoint, as it causes so much chaos. The easiest and probably most effective way to deal with that is to rework our fairy so that instead of the SCH commanding it to do things, the fairy skills would effectively be our own, but the fairy would be the focal point of the effect and animation. Everything the fairy does would have to be commanded manually, and all the ranges would be determined by the fairy. But the fairy wouldn't care about anything in itself. It wouldn't have problems interrupting animations or even doing simultaneous effects, such as giving an AOE heal while moving. The heal would simply originate at wherever the fairy was at the time the animation started. They would have to be able to be placed manually to satisfy high level SCH players, but I don't think that's much of an issue in the first place. And if all heals aside from the most basic ones turn into fairy skills, while the SCH itself become the source of damage and utility, then it would be pretty lore friendly, being the strategist/commander and the co-healer.

This would work well as the oGCD healer as well, as then most of the heals would be oGCD acts, and would preserve the whole pet dynamic perfectly.

The weird thing about SCH right now, is that it's a crappy shield healer. SCH basically just has Crit-lo to stand apart from the other two healers and their innate ability to shield/mitigate. Sacred Soil is just a Temperance/Collective Unconscious with more uptime at a greater cost. Regular Adlo is just an inferior Devine Benison/Noct Benefic. And Fey Illumination is just a joke from a mitigation standpoint when standing beside Sacred Soil. Somehow, SCH doesn't have any other shield/mitigation, despite being the game's preeminent shield healer.