I agree with this. Tetra is such a boring ability because lilies do the same exact thing and is a far more engaging system. It could be removed and WHM would fundamentally not change at all (it would actually be better since you'll have to deal with the consequences of improperly spending lilies). Same with Aquaveil, Exaltation, Krasis, etc. Buttons for the sake of being buttons.
People aren't complaining about wanting to do DPS instead of healing. The issue is that when there's nothing to heal, you have 1 button to spam and 1 button to press every 30s. It's boring.The fact is, healers press far more than 1/2 buttons NOW. Just the community has decided if they aren't GCDs, they don't count. So if healing was shifted to GCDs and oGCD heals were emergency/panic buttons (as they arguably should be), that would already fix the complaints...
...or reveal them as (some people) complaining because they want to DPS instead of heal and don't REALLY want to have to press more buttons if it means more healing buttons...
The only thing you've said before that I disagree with is that you want 1 healer sacrificed to be that mind-numbing 1 nuke 1 DoT job. BLM's core rotation is not the difficult part of playing BLM. RDM's core rotation is not difficult at all. Old SMN/SCH's core rotation can be considered slightly more difficult than the others but it still gave a lot of leeway. These design philosophies of what a caster can do should be applied to all healers because they're casters too. All tanks have a unique damage rotation that makes you have to think a bit, even WAR, so I don't really know why a healer should be kept like this.
Auto attacks becoming unpredictable would be a nice surprise, but the mechanics of every fight will still be following a strict schedule, so the abundance of oGCDS we have will still be overkill, at best slightly less overkill than what we have now.
My problem with unpredictable damage is that this design can only apply to future fights. If we adjust the next endgame fights to better fit healer kit design today, it'll still leave the entirety of ShB and half of EW to be snoozefests. I think it's more feasible that, instead of adjusting fight design to current healer design, we adjust healer design to current fight design. And even then, I think that a healer with a fleshed-out DPS rotation and less oGCDs will play far more engaging than current healers in an unpredictable boss fight.