Recognizing this as more a "what are your thoughts" than a "react to OP's thoughts" per se...
I adore the fundamental concept of the 4e skill system. Every skill is very very broad, intentionally and (IIRC) explicitly meant to cover a huge swathe of things that could never, and should never, be narrowly pre-defined. Arcana covers everything to do with arcane magic--you want to invent a ritual, disrupt a barrier, transfer a "cursed with awesome" curse, fool a detector, bind together a golem, etc.? Arcana has you covered. Even if rituals and the like already exist to definitively do some particular thing, the most relevant skill acts as a fallback for when that ritual isn't available.
At the same time, I appreciate the value of the 13th Age Background structure, where it is the player who decides what stuff they've learned to do. That's awesome, and helps integrate the character into the player's thoughts strongly. Of course, the problem is that Backgrounds can be both easy to make...not super useful on the one hand (if highly specific or esoteric), or ridiculously over-broad on the other (e.g. "Super Spy +5" because James Bond can do anything.)
Which makes me want to find a way to split the difference. E.g. everyone gets (say) 3 skills provided by their class, skills that are clear and specific and which can be broadly used. Paladin, say, might get Religion, Diplomacy, and Athletics. But then everyone also gets two player-defined Backgrounds, which give the same bonus as a skill training, but which apply to things worked out between the player and DM. That way, the player-chosen Backgrounds can be as weird or mundane as the player desires, so long as the DM is cool with it. Things that give bonus skills could then give both 1 extra skill training and one extra Background, allowing players to branch out if they so choose.
Skill Challenges were a great idea and should be brought back. They can be improved over their final 4e presentation (not surprising, it was a new idea), but the improvements are mostly flourishes and prying things open a little more than they were before, the fundamental structure is already solid and functional.