The mechanics of Lilly are a but to RNG compared to the other healers.
AST, they have 6 buff cards, and can even exchange a card they don't want for a healing card or a Damage card, they have a decent amount of flexibility to go with the flow.
SCH, has a good amount of support and a nice addition of new abilities that aid even further in DPS.
WHM lacks any of this, while I imagine your thought is to have WHM not have to DPS, you shoot yourself in the foot by saying let's focus on Cures and ignore Stone, Aero I/II, Aero III/IV and Holy. In a way by pretending those don't exist you limit the potential of what a WHM could do.
If you use these abilities you could do the following.
Mechanic 1: Elemental Synergy. After using an Earth, Wind or Holy based action you next curative ability will be augmented.
Reason: Because of the design of WHM to already heal by a lot, adding a mechanic in which they weave in Damage to augment their curative spells in some way gives WHM more depth compared to the RNG mechanic which you currently have planned for them.
Mechanic 2:A stance which puts a limiter on WHM's healing say 20% and as time passes the gauge slowly builds up while in combat. This mechanic could in a sense by like NIN's auto-attack buildup of their gauge except this is a gauge that builds up overtime as long as the WHM is willing to put a -20% healing potency on the Cures. With the build up gauge you could then add another ability which gives WHM's control over when the exhaust their gauge. The lack of an option when it comes to exhausting resources also hurts WHM, AST can hold a card, redraw cards and have decent flexibility with their cards, yet WHM, with 3 Lille's is forced to use them if they want to use one of their OGC abiltiies. A WHM may want to wait for the right moment to use the Lilly since they have 0 input on when to activate it other than when they choose the ability that uses said mechanic.
Reason: WHM's are already capable of overhealing with ease, so why create mechanics which just make their heals even more OP? The Lille doesn't create a scenario of an uh oh button because of the way Confession and Lille's work, Confessions are to RNG that you have to actively cure/cureII allies in your party and then hope you can keep them up in the 30 second duration all for a pitiful 300-400-500 potency AOE cure. Which isn't the best payoff when you consier the AOE utility other jobs have like Indomitability, which is also instant, 400 potency, AOE cure, that just uses aetherflow. Meanwhile for WHM to get 300 potency for a target they need to have at least 1 confession up, meaning that this instant ability is pretty much worthless because of the effort it takes to just get 300 potency on all allies.
Mechanic 3: Same as above but instead introduce an Exhaust ability which "upgrades" Stone, Aero I/II, Aero III/IV, Holy and Fluid Aura to have a buff effect on target ally. You can only use one buff every 15-30 seconds as the Exhaust wears off once you use said buff but this would allow WHM some form of buffing in the game that they need.
Reason: WHM need buffs that assist allies, It works for DPS jobs to not have buffs for the party because they bring their DPS to the game, healers on the otherhand as you have mentioned before, may just want to heal, but by only giving WHM the options to heal or DPS, you kill their opportunity to be of use to most groups because they provide 0 utility compared to SCH and AST.
Mechanic 4: Purify, like Esuna, but it takes an enfeeble and turns it into a Buff. An ally can only be Purified once every 30-60 seconds.
Reason: Utility, give WHM something to make them special, the ability to turn certain enfeebles into buffs would be a decent one, maybe have it where only the white bar enfeebles can be purified but then the thought is, do all the enfeebles get purified or only one? Honestly I"d say all, maybe have it where where for each enfeeble Purified it adds 30 seconds to the limit of when said person can be hit with purify again. So say you get hit by bad Breath, Purify it removing all applicable abilities that can be swapped to positive effect to positive effects while leaving the negative ones on.
So let's say, Blind become Crit rate up. Heavy becomes Sprint, Slow becomes Haste, Any attribute down becomes an attribute up and nothing else comes off the to of my head but the following couldn't be purified.
Paralysis, Bind, Amnesia, Silence, basically only enfeebles which reduce your attributes in some way could be purified.
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Those are the 4 that currently come off the top of my head, but basically, with WHM you need to add more utility to them, people don't care if WHM can overheal a tank by 400% because that 300% waste is just that, waste. Remember you called WHM/SCH/AST "Support" Roles, they aren't "Healer" Roles, they Support the party. Currently AST/SCH provide Support to the party, WHM doesn't and that's what you need to reevaluate when it comes to their mechanic and overall playstyle.