I really, really like this suggestion and it's something I've been looking into myself for a long time. I'm actually in the process of designing something similar (but not identical).
A stat-less RPG would definitely be fun to play. It will have some design challenges to overcome which will probably require some ingenuity and innovation.
The reason why stats are so prolific in games, especially lengthy ones like traditional RPGs, is that they fool the player into a false sense of progression and difficulty. A bunny you kill at level 1 is not much more difficult than a demon you kill at level 50 or whatever. The difficulty in the challenge lies solely in the difference of numbers but the execution (ie. the skill required by the player) remains the same: choose an attack that will efficiently deal lethal damage to the enemy.
Sure, some games do genuinely become tougher over time as enemies employ more complex tactics but it is those tactics that make the encounters difficult, not the numbers. A demon that only auto-attacks is easier to kill than a demon that knows how to flank. Given the same tactics, the demon and the bunny are no different in difficulty, stats not being a factor.
On the player's side, some sort of progress indicator still adds value because people do become more skilled at a task over time. But, in the real world, a skilled swordsman is better than a novice not because he can take in more blows (bigger HP bar) or deal 5 times more damage, its because he knows how to avoid taking lethal damage and can employ skills and tactics out of the reach of the novice. Therefore, an RPG of this nature should emphasize skill progression on level-up instead of attribute progression.
Going back to our demon, a low level player will fail against a demon because his reaction time, speed of execution, and limited skills and tactics are not sufficient to overcome the threat. A high level player, on the other hand, has an array of offensive and defensive skills at his disposal to handle the situation, couple with faster execution time and more stamina (not HP). However, even though the player is more skilled in-game, the encounter is still more difficult than taking on a simpler enemy at low levels because it requires the player to execute a more complex strategy, making the encounter truly more difficult instead of falsely difficult by just making the creature have bigger stats.
Now this does not exclude stats altogether. A bear is definitely stronger and heartier than a mouse. This approach should attempt to model relative strengths through more than stats alone (hidden or otherwise).
The downside to this approach, and the reason why it's probably not more prolific, is that there is a soft cap for how long the game can extend this concept. Traditional RPGs can last forever since it just boils down to making the next set of tougher creatures have bigger numbers and requiring the player to level up to the point where they can match said numbers. This system will cap on the player's ability to execute on complex strategies expeditiously with a razon-thin margin of error. That cap is variable per player, variable for the enjoyment of the player, and has a definitive ceiling.
