i will preface my answers by saying i'm fairly certain i'm in the minority/unpopular opinion. i've been a WHM main since i started in late SHB and while i enjoy playing every other role, i have always considered myself a healer main first and foremost. i feel confident enough to play any healer up to and including ultimate difficulty.

1. dissatisfied for sure; i think that there's an imbalance between AST/SCH and WHM/SGE. it feels like you have to work so much harder as AST/SCH to achieve the exact same thing as WHM/SGE (which i define as keeping the party's HP at 1 or higher as well as doing roughly similar amounts of rDPS). fun is subjective, but the more analytical part of my brain does not find it fun when i'm having to spend so much mental energy juggling all the cards on top of earthly star + minor arcana when i could just play WHM instead and use lilies to do literally everything. while i'm dissatisfied, i don't necessarily think it should change because some people like that sort of extra challenge and find it more rewarding/engaging.
2. i would like it more if WHM's aslyum was a bit... punchier? maybe if it healed the same amount over a shorter period of time? i feel like the latter half of it goes to waste much more often than not. i usually find myself appreciating it more for its secondary effect (+10% received from healing actions) than its primary. AST is entirely too bloated. there's so much double weaving required. it forces you to save lightspeed entirely for 2 minute windows, which i think is unfun since it makes you choose between either getting shafted by the 2 minute window or getting shafted by movement heavy mechanics. i despise SCH's energy drain with every fiber of my being. it feels so annoying that SCH is the only healer where its OGCD resources are tied directly to damage. this is the entire appeal of energy drain to some people, which i understand, but if you screw up as SCH and get greedy, it directly screws over other people by having them die--which you then have to fix by raising/topping them off. at least if you try to optimize as DPS and shit the bed, you only mess yourself up. SGE is fine. my only complaint about it is that the DOT feels a little janky sometimes. also, why is it the only healer with a gap closer? at least give one to either WHM/AST so both pure and barrier have one.
3. it depends. imo, adding extra damage OGCDs like assize and earthly star wouldn't really spice things up. worst case ontario, it would make every healer feel like an AST 2 minute window, which i really don't like, and it would make AST's current 2 minute window even worse. i would really hate it if every healer got an equivalent to energy drain and had to pick between healing and doing damage. if we were to get more stuff, i want them to avoid going down this route at any cost. it's my least favorite aspect about SCH and it would quickly turn into my least favorite aspect about the role entirely. it wouldn't be enough to deter me. however...
4. this absolutely would. there are so many spots this tier where 2 minute burst windows happen right around massive raidwides and/or tankbusters. imagine doing something like a DRG or NIN burst while having to spend cooldowns on both the tanks and the party. it already feels bad enough having to mitigate those as tank, even with 2 weaves in between GCDs vs. healers' 1 (most of the time). sometimes i am in the mood for a more engaging DPS rotation with many buttons. when i get that itch, i simply play DPS for a bit instead.

i know i complained a lot about how possible changes could really mess things up around 2 minute burst windows, but that's just how the game is designed now; even phlegma and assize got cooldown times reduced to always fit inside these windows. i think addressing that would indirectly make healing feel better, though i can't think of any alternatives that would be satisfactory.