I think the issue is less one of 'maths' and more one of 'feel'. We can look at the concrete maths of how many GCDs are spent on one thing or another, but maths can't explain everything. Take the WAR issue, by the end of SHB there were a fair few people that were getting concerned that, in EW, they'd add another button that the Fell Cleave button would turn into, so as to have even more 'press this button over and over'. Five Fell Cleaves, two Inner Chaos, potentially something else added to the mix, it was getting very same-y for a lot of players. Now we have 3 FCs every 60s instead of 5 every 90, which, for some, doesn't solve the issue, but for others, it does help to break the monotony a bit. I personally swing both ways on that one, I never liked 'press 5 FC' when it had no thought behind it, but I DID like 'press 6 FC, after spending the previous 90s setting up to have those 6 FC all land in your burst window', because then the ungabunga was a reward rather than just a freebie thrown my way just for being alive at the 90s mark
Back to healers, if we look at these two rotation charts I quickly mapped out (Stone is the Nuke, Dia is the short DOT, Aero is the 'long AOE DOT' as they removed it from the rotation maker. Did it this way because looking at so many Glare and Dia icons was getting hard to read and they were blurring together)
Replace any three 'Stone' casts with Solace/Rapture in that second one, based on where the healing is needed. I am not trying to pick fault with what you say, I agree that the numbers show that we cast roughly the same number of Non-Glare casts per minute as the past. The problem, I think, is that Rapture is doing a lot of heavy lifting and padding the metrics for the second picture. The problem to address, in my opinion, is not the 'number of times we press Nuke' overall, but the 'consecutive Nuke count' for each picture. In the first, the longest string of 'Stone' casts without interruption is 6. In the second, it's 12, before considering where to place Raptures. If, for whatever reason, you had to back-to-back three Rapture casts, that would still be 9 Glares in a row. Or you could ration them out and have it as exactly 4 per string, but damage doesn't really work that way in FFXIV raiding, it's pretty 'all or nothing'. And with stuff like Plenary, I find myself using two Raptures back to back, for that juicy 200 bonus potency per cast
All this is to say, it might really be as simple as just, 'bring back the SB rotation (the first pic), and put the new Lily system on top of it':
Again replace any 3 casts of 'Stone' with a Lily spend (provided you're using 3 before the next Misery). Now we've got the first pic's amount of non-Stone/Glares as a baseline to work off of, while also having the versatility of where to place Rapture/Solace as needed to heal with. I might be biased, but I can't really see how that'd be objectionable gameplay. Just have Aero 3 upgrade if we really want to get rid of every elemental theme that WHM has, we even Lost Banish as a 'target enemy, do AOE around them' spell in Bozja, with the VFX in the game ready to go.
I'd be okay with numbers like, going from 60/60 per tick on Dia (660p), to say 100/50 per tick for 18s (400p), Glare3 going from 310 back down to 300 (rescale all previous forms if needed, and Misery goes to 1200). This, according to my very quick maths, leaves us with this 'new' Aero 3 needing to do around 610 potency over it's duration, to keep WHM roughly equal with it's current potency output. 130/60 per tick puts it at 610, job done. But WHM's kinda behind on damage, so I'd propose the Banish upgrade bump that to 150p on cast, and 75 per tick, for a total of 750 over it's full duration, and see where things land. Reintroducing Cleric might accidentally make WHM way better than AST though. Might be ok if Cleric were once per 2min instead of 1min, but it'd be a lot less interesting a skill if that were the case