Non-DPS-aimed cards in the card system
I've got to wonder why you feel this is an issue in the first place that you have options to give someone extra damage reduction, a fairly strong HoT or a fairly potent shield (if nothing else use it on yourself for extra protection against the next Aoe?). Plus you can always not use them, compared to the previous system you're not beholding on spamming out your cards to fill Astrodyne after all.
But, assuming that you mind it on a principal level that you got cards of different mental value, wouldn't the correct solution be to remove the DPS cards entirely, and make all cards option, which in turn frees up design space for making their effects much stronger and supplanting other existing spells and oGCDs to make us more focused on cards? After all, the reason our card system always wobbles so much in each design pass is because there are DPS options in it, which automatically become the only ones that matter.
Feels to me like the issue is having DPS cards, not having non-DPS cards?
Weak buffs
Well, to a degree I agree. But it's more a problem with job design overall in this game, not Astro in particular. We have no Expedient, sure, but then again neither do White Mage and Sage. We have crazy stuff like Macrocosmos which if casted at the right moment gives a mobile GCD + an automatical heal of a major mechanic 15 seconds later (since you want to cast it so it goes off on its own). We also bring stronger damage buffs to the party than any other healer in our cards and Divination.
Could they be less, but stronger? For example there's no need to have 4 damage reduction buffs total instead of a single very strong or short-CD one? Sure. But again, that's an issue that virtually every job in this game has that they needlessly have 3-6 individually weak or long-CD buttons doing just about the same thing just to fill up hotbars and make the job feel way more complex than it actually is. Solve it at a game-level, not an Astro-level, IMO.
Busyness
Eeeeh... with this one I'm not sure what to say about it. I mean yes, but also, that's why you play Astro, yes. That's the one unique thing it has actually going for it, eshewing the veneer with the cards and all for a second. It's a busy healing job, with loads of individual bits and pieces, compared to the (comparatively) straightforward but powerful hitting White Mage. Astro has all the extra utility bits bolted on you keep firing off, in return you got all these utility effects to fire off in the first place.