Will try to keep it short cause I could write some pages on how some aspects could change and some ideas that i've said in the past:
Personally for its dps, lean more into its unique aspects like addersting management, the augmentation of skills via eukrasia-like actions and phlegma-like attacks in terms of range, spitballing some ideas:
-A CD based eukrasia action (kinda like the pvp one) that open new combos depending on what actions you press (i.e eukrasia II->Eukrasian dosis II that gives a buff that changes dyskrasia into a new finisher and Eukrasia II->Eukrasian dyskrasia II that gives a buff that changes dosis into a new finisher) both combos could have different range one being ranged and the other having the same range as phlegma, give different effects (one could give a damage buff while the other could have more potency and higher kardia heal potency) and both could generate addersting
-Eukrasian dyskrasia (just copy miasma 2) and Eukrasian toxicon must happen, in the case of toxicon we could have a mini combo i.e eukrasian toxicon becomes the instant one but buffs the next toxicon used that could have a standard cast time, allowing the current niche of mobility while allowing some short of rotation of E!toxicon->toxicon->E!toxicon as well as some resource banking for burst windows
-They could introduce new ways and cooldowns to generate/use addersting, for example the idea of rhizomata generating addersting is good if we had some alternative uses of that addersting that is not "dps neutral instant cast" and if they want the sage to be generating/spending dps resources as a semi dps they could add traits like the old quickened aetherflow so that by using some actions you lower the cd of others (i.e you lower the cd of phlegma for every addersting used, phlegma lowers the cd of the addersting generators and the addersting generators lower the cd of the addersting spenders)
For the healing part I do agree that Haima and panhaima are good, in fact they are the most unique thing sage has to offer but they are not the rule, they are the exception, sadly addersgall is the bread and butter of sage and its a carbon copy of Sch cooldowns and changing this would be too big for something that is not called a rework but they could enhance the sage identity in different ways:
-The idea of a partywide kardia is good but this doesn't need to be a new action, they could change soteria like this "Grants 4 stacks of Soteria to all party members each stack healing every time a Kardion heal is activated, cure potency: 120" that way it would fulfill its current niche of extra single target sustain but also work as aoe heal
-If its the dps healer pneuma being the only heal that damages makes no sense and limits optimal use of Zoe, they could rebalance physis 2 and holos to work like pneuma (preferably those 2 being ground targeted attacks so they can be used in downtime) and give a bigger emphasis on Zoe, they could even recycle quickened aetherflow once again and make it so every addersgall used lowers the cd of Zoe and those heal/attacks
-Pepsis should be reworked, something on the line of "heals nearby party members with a potency of 100, if they were under the effect of the user's Eukrasian prognosis, Eukrasian Diagnosis, haimaton or panhaimaton restores an additional hp of (insert shield potency), if this effect happens aditionally reduces damage taken by 5% for 15s" would do wonders at all levels of play
-Something so kardia can be triggered during downtime, maybe an action that disables kardia for 10s but automatically heals every 2.5-3s with a 70-80% of the kardia potency
These are some (spitballed and probably unbalanced) ideas but the reason why I call it soulless is simply because it has no identity, they call it the dps healer but its dps side is identical to the other 3 and offers no significant differentiation nor its kit is deep enough to be granted that title and if you look at its healing side things are not bright either when the bulk of its healing is made in an almost exact way Sch does. This can go so far that in P8s, one of the most heal-demanding fights so far in EW only 20% of my total healing were those unique actions (+pneuma) that you mention, being the remaining 80% actions that are carbon copies of Sch's like kerachole, ixochole or physis