Honestly, it's kind of impossible to tell what on Earth the design team actually wanted from the healer role out of the get-go, what they want of it now, and what they would like to do, but feel like they can't.
That said, this is my guess as to what they wanted.
When ARR released, they wanted healers to dance between offensive windows and healing windows. I infer this because all fights, even during ARR, dished out damage to the party and the tank infrequently rather than constantly which is what WoW did at the time. Yoshida and the design team spent a lot of time in WoW before redesigning FFXIV, so perhaps they felt the amount of healing required back them felt too demanding and/or maybe didn't feel fun to them to be healing for almost--if not the entire time.
This take was pretty new for healing at the time, and a lot of people coming from past MMO experience weren't expecting this. Going into mid-late ARR and into HW, the awareness of not only how much freedom healers had to add damage to fights, but how valuable this was started spreading, and Cleric Stance in particular became a frustrating experience for quite a few people who either weren't used to dancing between offense and healing, or who just weren't familiar enough with FFXIV yet.
Since then, I think the design team feels traumatized by that experience and are afraid that any amount of lateral movement back in that direction will backfire, and this bleeds into this idea that healers need to be coddled. And the coddling isn't just us not being allowed DPS buttons... It's constantly bloating our free healing tools with each expansion, making healing buttons stronger and stronger, removing mroe and more non-healing-non-DPS utility buttons like Disable, Virus, and Eye for an Eye, and refusing to repeat any old healer mechanics or add new ones that isn't just "each healer targeted by two stack markers" Remember Searing Wind? Remember Briny Mirror?
Of course, this is all just assumptions. That all said, healers that are also responsible for offensive duties is not bad design, nor is it exclusive to FFXIV. Healing-only healers aren't bad design either if done correctly, but the actual content doesn't really create an environment for that playstyle to thrive, and I think the design team can't really do that without creating a schism between new and old content, and also they probably fear that decision would drive even more people away than they already have, so they're feeling stuck in this sort of sterile middle ground where most aren't bothered enough to leave, but everyone wants more from the healers for different reasons. Some want DPS rotations, some want to remove the DPS requirements from healers, and no one can be 100% happy because making one camp happy means the other won't be, even if it compliments the way this game is designed.