Understandable on the note of longer cast times making immediate healing more difficult, but I imagine that, similar to black mage, white mage would then get other tools to help compensate for the fact. While not exactly giving them triple cast, it might be interesting if they had other tools to give them moments of speedier casting. If not a new button entirely, they could for instance give Thin Air 3 charges rather than two, make it reduce the MP cost of its next spell by 50% rather than 100%, and make it cast instantly like swift cast, for in the moment instant casting to compensate for the fact they would otherwise have to spend more time casting. I also feel that white mage, along with astro and scholar, should get some kind of mobility tool like sage's Icarus to be honest.
As far as the comparison between ast and whm go, I think the same can really be said about any healer really. They all need baseline healing enough to get through the content, and ast's strength came/comes from the efficacy of its healing. I mentioned before, but the idea is that where ast has powerful ogcd healing on cooldowns, whm would rely on their GCD healing, without falling behind in terms of damage. The idea isn't that whm would have more healing than the other healers, but instead that the gcd healing it provides is potent without forcing it to lose out on also dealing damage. In terms of late content like savage or ultimate, ast will always have an advantage over whm due to its cards and divination. Simply the fact that it provides bonus damage will always give it a place over the other healers, which is why I mentioned making something like that feel more impactful and a bigger part of its identity rather than just a tacked on gimmick to keep it busy. By comparison, whm would be extremely self-reliant. Its personal dps would be much higher, and the consistency of that damage far more reliable. Same with its healing. Other healers might have the potential to do more, but none of them would have the same safety and consistency of a whm. In the case of savage or what-not, it would mean a whm can cover a lot more of the required healing without forcing it to lose out on dps, making not only their job more reliable and keep it ahead in terms of personal dps, but also help the co-healer focus more on whatever its niche or identity is.