The problem here is you seem to be under the misconception that WHM heals are stronger.
Let's take your first example. AST's CU is 150 potency, and follows the party when they leave the bubble or the AST cancels (which an optimal AST will). Asylum is 100 potency. Already quite a bit weaker.
A better analogy to SCH would be Whispering Dawn, which is 120 potency over a similar duration. Closer, but still more powerful. Benefic 2 is 715 or 747.5 potency, vs cure 2's 700 potency.
Now we can go on and on like this comparing potencies, but in the end I don't think it's what we should be looking at. At the end of the day, all healers have powerful heals. And I don't think anyone thinks healing in this game needs to be more powerful, HP bars in high end content yo-yo enough as is.
So then we look at other things.
MP? Well WHM was stronger here until AST was buffed, so it no longer retains the advantage.
Enmity? AST generates quite a bit less of it, both through stances and things like ES technically being a pet ability.
Utility/raid buffs? Well, once upon a time WHM had an upgraded protect and things like divine seal, but all of these were stripped and shared with the other healers. So now if you want unique utility, you must pick AST or SCH.
Damage? WHM did have an advantage here, but SE buffed AST to be above WHM and SCH has been on top since the start.
Mobility? AST is the clear winner here again, with the DPS spells getting reduced cast times, and Lightspeed getting CDR and buffed to synergize with DPS spells better.
SCH also gets an extremely powerful class mechanic - a whopping 480 potency oGCD regen, compared to WHM maxing out at 150 potency regen on a single target. And AST's class mechanic is, of course, cards, which can directly provide extra healing or damage as well as a number of raid buffs. Gated by RNG, but that RNG has been buffed itself many times to play nicer.
So, what exactly does WHM bring that no one else does...?
*crickets*
What does WHM bring that's stronger than anyone else?
Well, cure 3, sort of. It's 50 potency stronger than Indom, but quite a bit weaker than Earthly Star. Earthly Star is also oGCD, costs no mana, and generates no enmity (while cure 3 is notorious for ripping hate from tanks). Buuuut... cure 3 can be spammed. Except nothing in the game requires it. Any time cure 3 might have come in handy, Indom + ES can clean up handily.
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WHM could be flat buffed quite easily and not become OP. Exactly how to do that is open to debate. Personally I'd be happy with Balance and Chain Stratagem as Role Action, but I doubt SE will do that. This thread is more themed around buffing WHM DPS. It's a very practical solution: AST + SCH can bring the raid buffs and still be desired for parse runs like always. But if WHM damage was not crippled, potentially offering raids higher DPS output than the other healers (instead of lower) could be a bit of a consolation to running a non-meta build - instead of feeling like AST is a flat upgrade in every way.
Long term fixes are much thornier. As long as SE insists that WHM is a "pure" healer, but not a strong(er) healer, and can't have utility, competitive DPS, raid buffs, and so on - and considering that most of AST's healing is a reskin of WHM spells - we're going to go through this cycle again and again. But WHM needs an advantage somewhere.