I think that it should always have the greatest role possible, except in tabletop RPGs. The one reason I prefer Oblivion over Morrowind is that in Oblivion player skill mattered, whilst in Morrowind it was all about dice rolls. However, I wouldn't like an RPG if some sort of limitation around skill based on in-game stats wasn't in place. A level 1 character should never be able to defeat a level 10.
In most RPGs, I tend to think of stats as the way the game blocks off areas. Whilst in most games the game will block off an area until you progress there by beating previous levels, in most RPGs, even the highly linear ones, there will be areas you can go to, but not progress in, simply because you don't aren't good enough.
Another important thing is that an RPG forces you to strategise, rather than just have fast fingers. It has to make you think about your skillset, remember your stats, and be able to make a fast decision. Note how I say a fast decision, not a fast action. By this, I mean battle-changing stuff, like running away, changing your position, or gulping down a health potion. Not an action that will live on for a few seconds before becoming unimportant, like attacking EVIL GOBLIN WITH POINTY STICK Nº1.
The only RPG I've ever played that I think did it perfectly is a rather hidden gem called Risen, made by the some of the people that made Gothic 1 and 2 (Which I've never played). Understanding of the combat system and being really, really good at it allows you to go to areas meant for much later in the game, but there are still areas you just will not be able to go to at level 1, because the enemies are just too strong. If you had a lot of trouble you could grind up to make your life easier, but ultimately, your progress through the game was dependent on your skill, not your character's. It didn't force you to think much, though. The only two "Battle changing decisions" were to gulp down a potion or to switch from melee to range, or viceversa.