Huh, this isn't a bad system at a glance. I can really appreciate how you've put an emphasis on growing from failed rolls... however, I think the exact numbers could use a little bit of work. Speaking here as a long-time veteran of Pathfinder, DnD 4e and 5e, Mutants and Masterminds 3e, Savage Worlds, and a couple other systems, there can be a lot of rolling in just a single session. Many of those, particularly with your high 'average' DC of 15, will be failures. That's a ton of skill growth per session, and I'm not entirely sure this system would be at its best weighted down by the number bloat that will inevitably follow it.
You've taken a lot of care to keep this extremely simple, which I can respect, but constant leveling without much in the way of a limit - and with accelerated speed on failed rolls - could lead to adding giant numbers to dice rolls in short order. From experience, that tends to make things trickier for people fresh to the tabletop. Further, I'm of the opinion that being able to advance all your skills so quickly from doing poorly weakens the dynamics of a party; where typically most people in a group fill a unique roll or three.
I think to adjust that and still keep things simple you can either: change the rate of growth in skills (so you don't level up from 1 to 2 by a single failure), or use a 'proficiency' system so you pick a few skills at character generation that will grow from failure as your character is particularly bent to learn from mistakes in that area of their life (a typical barbarian probably doesn't care to grow their arcane stat even if they roll it reluctantly or as a joke). I'm personally a fan of the latter method, because it will help your party fill their unique niches and doesn't stray far from examples set in other systems.
Besides that, I think there should be clearer limits on things like out-of-combat magic and perhaps some sort of level cap so gamemasters don't have to continually escalate threats to challenge a party over a longer campaign, but that's more of a personal preference thing and down to what you want from your creation, here. I hope this all helps. ^^