I'm not attached to stats, and I think games can do just fine without them. Too often they're taken for granted as a must-have without a lot of thought regarding what they actually mean within the system. As marginal bonuses in arbitrary categories, they can add unnecessary complication, and if they're just really broad skills, that calls into question the need for them at all. There's a reasonable case for them in truly combinative systems (white wolf, cortex) but even then they can just as easily be swapped out for almost anything else.
This is not to say stats are bad, just that they are an unexamined habit. The question should not be whether you exclude them from your game design, but rather why you feel you need to include them.
LUG's Aria was a brilliant and nearly incomprehensible game that did something clever with it's stats. There were something like 12 stats, but you only had ratings in 10 of them. the other two you were just assumed to be normal, neither good nor bad, and they didn't come up. This struck a spark with me that ended up influencing the design of aspects. One thing they were going to replace was stats, with the idea being that your stats, whatever they were, were all normal unless you explicitly called them out with an aspect like "Strong" or "Slow". I admit this is about the sweet spot for me, but as with all things that are player defined, it has some complications.
Stats are incredibly important to the old Amber DRPG, in that they are really the heart of the character. There are 4 stats (Strength, Endurance, Warfare and Psyche) and each is effectively a super-skill, with all conceivable activities covered between them (unless those abilities are explicitly in the powers system). The ways in which this system was broken would probably make its own post, but that worked to the game's advantage because it encouraged homebrewing, which is a robust part of the ADRPG culture.
So while not-amber should absolutely not include those four stats, the four stat model itself works very nicely, so the questionis what the stats should be. There's an instinct to fix the ADRPG list, but that seems overly derivative, and it fails to address the underlying issues. Thankfully, I already had to solve this problem for the Road to Amber MUSH and I am more than content to steal from myself.
The trick with the RTA stats is that they do not have any influence on what you can do, and only tangentially speak to how well you can do it. Characters in this context are assumed to be broadly competent and unless a particular task is constrained by system (like magic) or common sense (like flapping your arms to fly) then it's assumed they can go about it competently. With that in mind, the RTA stats speak to how you do things.
The stats are Force, Grace, Wits and Resolve and it's entirely possible to use any stat in any situation, depending upon how you describe doing it. Swordfighting, for example, is easy to break down. Do you go in swinging?(force). Do you dance around nimbly? (Grace). Do you wait for your opponent to tire or make a mistake? (resolve) Do you study your opponentes moves, lookign for an opening ?(wits)
Certainly, some actions gravitate more towards some stats than others. It's hard to read a book with force, but then, it is rare that reading a book merits bringing in stats. If the book is so exceptional as to merit that, there's a good chance the exceptional circumstances also invite broader stat uses.
Now, this is implicitly a combinative system, in that when you enter a contest, each side chooses the stat they're using, and each sides's effectiveness is determined by combining the two stats in question (So if I use force and you use grace, the winner is decided by the higher force+grace). In MUSH, this makes for a nice guessing game, as you want to use your strong stat, but you might be incented to use a stat you think your opponent is weak in.
Translated to the tabletop, this changes a bit, at least in part because challenges will not be with other players (at least as often), but may be against the environment or NPCs, which is to say against challenges set by the GM. Provided that there's some way to handle blind betting (so to speak) when appropriate, the system works just fine with inanimate challenges and NPCs by t GM simply assigning them stat equivilancy. That door might have a high Resolve and Force, but negligible grace and wits (it's a door after all) and it always uses Resolve (because, again, door). That mean breaking it down (force & resolve) is harder than picking the lock (Wits & resolve) but either approach is doable.
Ok, don't take this as cast in stone, but let me illustrate this with a sample system.
If I was inclined to express the stats as dice, I' go with something influenced by the new designs in the Cortex system. Normal/Negiligble is d4, with values going up to d12. Fast and dirty chargen is to give 4 d8s (or d6s, depending) with the rule that you can drop a die one step to increase another die by one step. Minimum d2 (indicating an active problem with the stat), maximum d12.
Combine that with the Baselines from earlier in the week and you quickly get a character sheet that perhaps looks like:
Scion of Argent: d6
It's still not quite there, but the bones are starting to emerge.
1 - From that point we realized th emechanic could just as easily handle "Drunk" and then we were off to the races
2 - It is genuinely a terrible game, but only in the same way that raw sirloin makes a terrible meal.
3 - There are other elements that enter into this in the actual game, but as an abstract, that will suffice.