I've never really associated White Mage as the 'buff character' across all FF games. It certainly has those aspects (ProShell), but it's never been the distinguishing factor the way it has been on classes like Bard or Dancer, or even within the mage category, Scholar, Green Mage, and Red Mage. And that's mostly due to Faith/Bravery being very late additions to the series in particular while those effects also have been traded across classes as much as they were across the White & Black magic Spell Lists themselves. Support Magic often fills out lower level spell lists that multiple classes have access to across the entire series, and sometimes lean on for their identities through secondary mechanics that White Mage itself doesn't get (hence the RDM/SCH association). I do associate White Mage with strong healing and strong caster damage at endgame (thank you Holy), but if I were to give it a unique identity that fits that theme, it would probably be through something like the Confession mechanic we got in the later half of Stormblood, but way more involved, where you get this additional healing resource that changes the context of your spells (Lily and non-Lily alike), and enables you to DPS in situations the other two straight up wouldn't be able to.
As an example of what I mean, here's a theoretical rework of Assize that uses Confession in the manner I'm talking about:
Assize: Deals damage with a potency of 400 to all nearby enemies. Additional Effects: Restores 5% of MP and applies two stacks of Confession to self and all allies in range.
Confession Effect: Upon receiving HP recovery or regeneration from the next Cure, Cure II, Medica, Tetragrammaton, Afflatus Solace, Afflatus Rapture, Regen, or Medica II cast by self, Confession consumes a stack for an additional heal. Potency: 200. Maximum Stacks: 4.
Cooldown 45s. Maximum Charges: 2.
Of course, the problem with "300 potency regen ticks*" and "Do I cast Medica or Cure III here?" is despite their theoretical ramifications SE still has to design fights that strain those resources mid-fight, not at the last minute pre-enrage. A rework like that needs even all healers to at least be using healing GCDs 20-35% of the time in end of tier speedclears to feel impactful. And I think that's why something like the above Assize rework just doesn't matter (and why I haven't bothered posting my full reworks of all three healers), even if it would be fun to juggle in theory. If SE isn't going to put some effort into easing off their item level scaling and amping up sustained damage accordingly, adjustments to any of the healer's kits in the name of chasing raw potency will consistently fall into that trap.
That's at least partially why I advocate for squishing the item level gaps we have each tier and want to add a significant healing potency increase to Piety. Those are vectors that directly target that problem, and also have the benefit of making gear and 'dead on arrival' content matter more, without actually requiring SE to change much on their end. If the gap between min-ilvl and BiS healing requirements is smaller, then actual skill expression matters more, or at the very least, you won't see Succor becoming irrelevant within a month.