Interesting, this is basically what I was discussing in a bit in the big thread on page 1 about O5N. I didn't have clear-cut ideas, but the point about Assize/Aetherflow stacks not having a lot of good elements of choice involved with them was a point I particularly brought up because it resonates with me.
That said, I'm not exactly sold on the idea of "casting deeps makes you have to heal less" (your suggestions about Stone spells adding shields to the tank, for instance), since I feel like healers already have plenty of tools at their disposal to weave the two (oGCDs in particular). DPS combos are a little more intriguing but the healer playerbase already seems taxed to the limit mentally just at upkeeping DoTs, if the furor on the forums is to be believed. I think the best option would be something based around resource consumption like SCH stacks, where you have meaningful choices on whether to use it for healing or for DPS (Energy Drain could use some more damage oomph wrt that though). That way healers who can barely keep up with the DPS moves we have now can still use those resources for healing (revamp WHM lilies, make up something like Minor Arcana for AST but not eat your card draws, etc) but healers who push the envelope on optimization will have meaningful choices to make about whether they use their moves to heal or do damage.
Revelation Online, though a garbo game in a lot of rights, actually had a great hybrid class that was built around this philosophy - every one of Occultist's moves could be toggled to heal or do damage, but shared a cooldown so you had to use them appropriately depending on the situation (did you want to HoT the tank, or DoT the boss? decisions decisions). WoW's Discipline Priest is another healer that produces decent output relative to other healers but has a significantly higher skillcap, which could also be something that we mimic here in FFXIV. People always talk about "if a healer is a lot harder to get the same output as other healers no one will play them" but tbh people play Disc despite the extreme mental overhead relative to other healers (Mistweaver Monk is another one that is suitably more complex than say, Restoration Shaman/Holy Priest). A lot of people tried to make 3.0 AST work as well, the problem was just that it was tuned poorly and the devs chose to make a myriad of buffs to it thoughtlessly hoping that something would stick, and that in turn made it basically as easy as any other healer to play, definitely easier than SCH now.
Finally, to round out my thoughts on this discussion, I am really, really pessimistic that people aren't DPSing while healing in this game because "it's too hard to do both". They're not avoiding it because the game makes it too hard on them, they're either a.) new/used to other MMOs where the healing requirements are much stricter and once the other shoe fails to drop with regards to healing they'll start DPSing more, b.) just kinda bad overall and play healer because they can skate by without doing DPS which = less buttons they have to pay attention to and odds of them being flamed over this are fairly low, or c.) they just plain don't like to feel like a damage mage instead of a healer and resist doing damage out of stubbornness (not unheard of I'd imagine, but definitely not as common as the forums/Reddit would have you believe). Making healer DPS easier won't change this - the devs tried this with the removal of Cleric Stance/DoT prune, and we're still seeing people losing their minds over this debate. I hate to sound fatalistic but at this point the only thing that MIIIIIIIGHT impact the community's divide on this is a massive sweeping balance change that push healers so far in one direction or another that the "choice" of DPS is solved for them. And I can definitely see why the devs are so hesitant to take a firm stance on this extremely hot-button wedge issue.