First and foremost, all healers are damage-focused healers. This is not a SGE thing; this is a FFXIV thing. This cannot be changed without literally redesigning the game's encounter design from the ground up and remaking every single fight in the game up to this point. It's like wanting to tear down the load-baring wall in your home. You can't do it or the house will literally collapse on top of you. You'd need to spend an exorbitant amount of money reworking the house's support structure to take down that wall.
That is just how this game is built. No amount of kicking or screaming is going to change that. Even with how neutered the healers are, DPSing is still the dominant metric of healer viability, and is still what healers spend almost, if not entirely all their time casting with the GCD. The only way to effectively move forward with healer design and give them proper identity is to acknowledge that and plan around it rather than trying to ignore it barrel through it by any means necessary. The road is curvy, but SE is driving the healer car in a straight line ignoring every curve and constantly driving over the curb--whatever curb is in the way of "pure healing."
By accepting the nature of how healers are designed to function in this game, we can start to actually address their GCD gameplay to have identity which none of the healers currently have in this regard outside of very occasional uses of Afflatus Misery, Phlegma, and to a lesser extent, Ruin II. You can even find alternative ways for healers to apply their damage rather than directly attacking enemies. The lilies is a very simple example of this, but more can absolutely be done and it doesn't necessarily have to be locked behind healing either. Other forms of support, utility, or unique mechanics can also accomplish similar solutions.
Overall damage output or overall healing output are not metrics that can easily offer an established job identity. Having higher healing output has never really mattered because all healers are expected to be able to heal through any fight. Having higher damage can get really dangerous in terms of balance if one healer starts dwarfing the other healers and pushing them out of viability for high end content. That said, there are some possible exceptions. AST's Macrocosmos shows how highly potent healing can substantially make specific mechanics easier, however, not all fights have mechanics like that and it can still be healed up by any other comp of healers even if it's more challenging to do so. BLM's relationship with SMN and RDM has generally felt healthy for a lot of players being the one caster who cannot raise or offer any other forms of utility outside of Addle which all 3 have anyway, and thus if offers a much higher damage threshold. There's less room for error for the party, but the reward is a notably faster clear. SGE could follow in this path theoretically, offering no other forms of utility outside of its normal healing output and necessary mitigation, but instead being a much heavier DPS healer. The trade off becomes whether you want more damage or more room for error. On the topic of Utility, this is where it's much easier to give each job a more defined identity. Expedient has been quite a blessing for SCH, offering it a very unique tool for increasing your team's margin for error on movement-heavy mechanics.