I don't think the artha system of BW is doing what you say here. Artha is a largely metagame framework.It lets the player (and thus their PC) try harder than normal (spending Persona to add dice) and/or to get lucky (spending Fate to open up "6"s). But I don't see how it is part of the system for narrating consequences on a failure.
Prince Valiant, for instance, doesn't have a failure system; but its action resolution works fine treated very similarly to TB2e.
Over the Edge also has abilities that reflect a PC's skill, and in the 20th anniversary edition Tweet encourages adopting "fail forward"/"no whiffing".
I don't know what RPGs you have in mind as "coming after" and not using skills to represent skill. I've already mentioned 4e D&D. Marvel Heroic RP has some "fail forward" inclinations, and it uses ratings to reflect ability.
Ok, I think I now might have a way to express the concept I am getting at.
For a given part of a game mechanic there are at least 3 things that can be interesting to look at: What the game claim the mechanic
represents in the fiction, what the mechanics actually
does and what
functions the mechanic has in the game. The first and second of these are what we typically find in the game text. The designer usually has a pretty good idea what function these are supposed to serve, but relatively rarely write that out except for certain key mechanics. We can generally make a good guess on the function based on what the rule does, but actual play is sometimes needed (a lot of the purpose of testing is to check if a mechanic is actually serving the function the designer expect it to)
Take strength in D&D5ed for instance. It is claimed to
represent "measuring physical power". What it
does is providing a modifier on checks the humans involved associate with the representation, affecting the probability of
succeeding at a tasks.
If I were to guess at the
functions of this mechanics I would suggest at least these 3:
1) A persistent characteristic of a character, informing the success rate of certain tasks over time (Informs the "simulation")
2) A player motivator to have the player in with high strength character engage in more strength related tasks/challenges. (Individual behavior priming)
3) A in-group role differentiator, giving the player with high strength character increased spotlight in situations where strength is beneficial. (Group behavior priming)
One thing to note is that these match up quite well with the fiction. We would expect a strong person to succeed more in strength related tasks, be more willing to engage in them, and step up to the task if the group need strength.
Now imagine a fictional game that is exactly like D&D5ed except the following: When declaring an action, both task and intent must be specified. If you roll equal or over dc, task succeed and intent happen. If you roll less than DC the GM has to narrate something really dramatic that brings the story forward. GM are in this case free to decide if the task succeed or not, but the intent at least isn't fully acheived.
If we now look at the mechanic of strength, we still have the same claim regarding what it
represents. But what it
does has changed. It no longer (purely) affect task success probability. Indeed it more strongly correlates with intent achievement. It also strongly affects the probability of dramatic stuff happening in situations where strength is involved.
How does this affect the functions? Depending on the motivations of the group 2 and 3 might still be relevant, but for groups that really seek mayhem and drama it could have the oposite effect! And the first function is almost completely obliterated, as a weak character that inspires the GM to think of good succeed with complications might very well succeed more often in strength tasks than a high strength character that inspires the GM to think of more failure with twist scenarios.
This is where my claim about FF/intent on success games come from. The most likely function I can see for strength in this scenario (also informed by how people talk about how these games play) is as:
1) A narrative marker indicating what the player want the perceived value of the characters attempts to use strength should have in the fiction (Informs the narrative)
I think this still is still reasonably associated with the given in-fiction representation, especially when genre conventions is taken into account. However a trad player is used to the representation of physical strength to be used for a different set of game purposes.
This is why I think it can be instructive at least in a transition period to not think of the skill/strength in terms of inate ability (due to the associations with "wrong" game functions), and rather think of them in terms of the function itself.
I think this analysis while not perfectly matching any existing games, nonetheless is instructive in where actual differences lie.