I think part of the problem is that the dungeon has to be "generic" enough for players to get through with any combination of classes. You can't put in things that need a specific skill if it's possible that nobody will have that skill available.
On the other hand, solo job quests can and should be designed to require use of a specific skill set.
That sounds fantastic. Were they in 1.0? If not then I failed to pick up on the Sleep tutorial because I've got BLM to 50 and I admit I've never used the spell.
The way the game works now, with gifting you your new abilities at the end of each job quest, is basically the opposite of what you actually need for learning how to use your new skills. Occasionally they work around it by giving you a short first quest basically to give you the skill for the real task (eg. Rogue quest introducing stealth ability) and those are great. There needs to be more of that.
I think if you're not forced to use those new skills, it can be a natural instinct to fall back to the other ones that have served perfectly well thus far, instead of trying to use this new one that you don't understand. (Also I think some of the tooltips aren't that well written and seem confusing when the actual use is quite simple. I do read them all but sometimes need to go over them a few times to actually understand what they're saying.)
The Lv50 healer quests are the only ones I can think of that really make you put some effort into exploring your other skills to try to keep your NPC companion alive.
Completing the duty the best that I can is what matters to me. I do press more than one button. I perform as best as I can, I try to make use of all my skills, and that's good enough for story content - but I go pretty awful in the few extreme-level challenges I've tried so far.
Reading through all the topics on this board - not just this one, but all the other ones where the definition of a 'bad player' seems to slide from "someone who doesn't know their basic skills and/or isn't trying" to "someone who clears the dungeon in 30 minutes instead of 20" and it's disheartening. (Arguably a reason for them to bring in a parse/rating/whatever, because I don't know if I'm good or bad compared to the 'average' player or what standard I should be held to, but I hate to think I'm failing and not be able to know it.)
The other thing is, how do you give the healer more to do? Honest question, because I don't know what you're picturing. What is there other than making enemies hit harder so the healer has more work?
And in the case of Scholar, it could be just that they haven't been properly balanced to function in a low-level dungeon. If they can only get away with it up to the low 30s, that's the point at which Scholar properly exists and isn't being modified for a level below their 'starting point'. Plus I imagine it depends on the skill level of the rest of the party.
Anyway the correct answer is, not needing to heal doesn't mean you can go AFK but gives you more time to support the party by DPSing, casting Esuna, etc. and keeping an eye on everyone's HP. Things can always go wrong.