frequent use of inspiration for ideals/bonds/flaws, maybe compel those things to force actions if they refuse the DM gets inspiration to spend on the npc's/monsters
If you want to use compels, then you would compel the player based on the PCs bonds or flaws, and give inspiration in return.
I'm looking for some way to have character aspects have a bigger effect on the mechanics of the game than they do with just Background effect. I guess similar to 13th age or maybe even Cortex Plus, with concepts of character having not necessarily baked in mechanical effects, but the ability to give them weight beyond just (here's inspiration). I could graft on aspects and such to 5e, but I'm not sure how to apply them without breaking the already fragile game math.
How fragile is the maths? I'm not sure.
Anyway, here's an easy suggestion: allow Inspiration to stockpile, and then frame the PCs into situations in which they have to use their Inspiration to survive. (The non-stacking of advantage means that they can't use all their Inspiration on a single roll.)
Going beyond inspiration, for out-of-combat resolution you want a mechanic that introduces finality. So you need "let it ride" (ie no retries) for stat/skill checks. (
Here is a link to some 4e-era advice in relation to this on the WotC site.)
And you also need a non-combat scene resolution mechanic. Marvel Heroic RP, HeroWars/Quest, Burning Wheel and 4e all have this. 13th Age doesn't, which is a significant strike against its claim to be an "indie" version of D&D. The easiest to adapt would either be 4e's skill challenges (but using 5e principles for setting DCs), or HeroQuest revised's extended contest rules, which use a d20 and target numbers roughly in the range of 5e target numbers.
You also need fail forward, discussed above. The best treatment of that (as Jonathan Tweet acknowledges in both 13th Age and the 20th Anniversary edition of Over the Edge) is by Luke Crane in Burning Wheel. The key is to distinguish "intent" and "task". If the player succeeds in the check, the PC succeeds at the task and achieves what s/he intended. If the player fails, the GM narrates the consequences, but focusing more heavily on failure of intent than failure of task.