So, the way I'd handle this is to look at SCH currently. It (and SGE) can 'burst heal', and have regens, but they have opportunity costs. Aetherflow/Addersgall costs, CDs, Pneuma's strong but it's a 2min CD. So, WHM should be the mirror to SCH, where it has some mit effects, but with an opportunity cost: Lilies. Just add Stoneskin (and an AOE version) as Lily spenders. This sounds like the dreaded 'homogenization' sure, but consider what we get from doing this: SCH can put mits and shields with it's OGCDs, and instead of having to top up with a wasteful GCD of Adlo/Succor, they can now rely on the coheal WHM to throw out a 'damage neutral' Stoneskin2. WHM now has a way to get value from what is normally 'overhealing to prep Misery', by using a shield instead of a wasteful heal.
Additionally, and this is the big one, assuming all four healers have access to shields, we can have 'shield checks'. Consider the HP-to-1 of the new trial, or Seat of Sacrifice. All four healers have enough tools to 'solve' that check, and so it's possible to put that check in as a mechanic. However, not every healer has access to shields, so we can't have a 'shield check'. In UWU, when it first released, not everyone had Arm's Length. The tanks had PLD at 3min, WAR at 'you have to use Holmgang for it', and DRK at 'use Plunge to override the KB, and it also has Benediction levels of input delay'. BRD and MCH had nothing. So, Vulcan Burst (the small Ifrit AOE blast) was not just a funny meme where you press Arms and ignore it. It was a 'shield check', making sure you have a Succor/NoctAST one applied. So, imagine this: a new kind of bar, like the flashing red one for 'interrupt this', perhaps green with a shield icon over it. This indicates that there is an aspect of the move, that can be reduced or neutralized by having enough shielding to make it deal 0 damage.
An easy example: A raidwide with a bleed, and Infirmity (healing down debuff) for the same duration. The raidwide does two instances of damage, one frame after the other. First an exactly 2000 damage raidwide of Unique-type damage, and then the actual raidwide of however much. The 2000 goes off first, and applies Infirmity if it goes through your shield (or you don't have one), the actual raidwide then does it's effect. So you can choose to either shield through it to prevent the Infirmity debuff, or later on you can make the executive decision that 'okay we can ignore this bleed until after the Infirmity wears off, then deal with the damage via OGCDs' or such. We also had Photon in A11S, which set your HP to 1, and you were ticking from a constant DOT because the floor is on fire, so you needed a shield or you'd die to the next DOT tick. The issue is, it's now so easy to get an OGCD shield from a WAR, PLD, DNC, wherever, that it's hard to say if this would actually change anything, or if it would just be another 'ok we use Shake here problem solved'
I see others bring up the AST vs WHM comparison and that brings me to the point: it's once again WHM that got saddled with the dev's twisted idea of 'lets have a healer that focuses on being 'the pure healer'', we had it in SB and it was a colossal failure, we have it now and it's, admittedly less of a failure, but still a failure. It's getting tiring watching them step on the rakes over and over like Sideshow Bob. But at least Bob stepped on a different rake each time, this 'pure healer' fixation is the same rake again
edit:
When they first announced the split, my thinking was that SCH would have everything like Feint, Addle, Disable, Eye4Eye, etc, and be an absolute machine of buffing and debuffing for it's mitigation. Like, 'mitigate this 120% raidwide to a manageable level' would be 90% the SCH/SGE's job. Instead we have them able to outHPS the 'pure healers' because mitigation is just 'better' in like, almost every situation (and mit is considered as HPS by logs now)
Haste would be fine for a BLM, and I've heard Monk can play some bizarro SKS build, as long as you understand Optimal Drift and can keep Demolish lined up with ROF correctly using it