Though I don't main heal now, I did play SCH in 2.0 and AST in 3.0. I do have to agree that healers lack a sense of style that is different from each other, where AST was obviously designed to fill either Shields or Heals. Though each healer should heal and enhance a party, how they do this should differ from one other giving players a choice in playstyle within the role.
WHM has always been the direct healer with casting Cure spells. The problem SE only wanted it to be a pure healer class which isn't working with how the game is designed. They need to add utility spells the would maybe relate the CNJ's stone, aero, water spells. It's would be great to see a group wide stoneskin that would increase party defense rather than absorb damage. Maybe a Determination or Tenacity buff? Then for damage you could change it up with the WHM casting spells like Radiant Shield minus debuff or EnHoly the deals additional damage from a players attacks. I guess the tagline would be a more direct approach to healer; directly heal, directly buff.
AST I feel needs to focus on one form of healing and since SCH is shield, AST should be Heal over Time or HoT and it makes sense if you look at it. AST is a gambler job so having their main healing ability be over time, it frees them up to focus on card play. I played COR in FFXI and since it was basically Blackjack buff, it was nice to have extra time to maximize the benefits. You can still have Sect dancing in job but maybe make it more related to cards and damage related. I think in the past someone mentioned that Diural Sect was a buff stance and Nocturnal Sect was a debuff stance. Spells like Lightspeed could increase the speed of HoT rather than speed of casting?
SCH should try to distance itself from ACN/SMN, maybe a trait that changes Miasma, Bio, Shadowflare from DoT to direct damage / debuff? Also focus on pet and books.