Exactly what sort of skills RPGs should require from the player is a difficult question to answer (one of the reasons the definition of the genre has become nebulous as of late), though the general premise does lean towards the ability to think on your feet and work towards some sort of long-term strategy. Essentially, not so much about agile fingers as it is about having an agile mind.
The unfortunate problem is that many games opted to go with more stat-based approaches to facilitate this for the longest time, leading to the idea that more stats will solve everything; in other words, making grinding a little too useful. This can also lead to stat-inflation issues, which can utterly decimate any challenge an RPG had if the player over-levels. Only in recent years have they started to work towards something a little bit better. However, the next issue is that sometimes they stray a bit too close towards action-oriented games; leading to decent (if not great) action games with strong RPG flavours and good storytelling, which has led to some alienation toward players who were looking for a more intellectual approach.
Over-correcting is a pretty common error when you think about it, and the tendency for players to over-exaggerate their responses to a game isn't exactly helping the developpers get the feedback they need.
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Based on what I've played, some games have struck a fairly good balance:
Dark Souls
Much of the gameplay in Dark Souls is taken from action games; but the slower pace of the combat, the need for constant resource management, and unforgiving treatment for any errors made by the player demand that he or she always be thinking about the fight. There's also a heavy emphasis on learning how to read the enemy's attacks, looking for little tells which will warn you about which attack is coming. Watching the game without any context gives one the impression that it's a fairly simple and slow-paced action game; but all that isn't seen, all the thinking which goes into the player's actions, is what makes it an RPG.
Final Fantasy XIII
This one might seem a little left-field, but the combat in FFXIII is a curious anomaly. Auto-battle, especially early on, seems to disconnect the player from the party's actions; however, it becomes increasingly fast-paced as the game progresses and the player is then forced to focus on higher-level strategies (making the auto-battle function a necessity). The system still takes far too long to fully open up (which is another issue entirely), but presents a surprisingly amount of depth once you finally get there. The other interesting thing about the combat is the stagger meter, which quickly becomes critical to success in combat; it's more or less required to be filled or else battles will take forever, effectively making balanced setups surprisingly undesirable for once. The player is constantly forced to choose between defensive paradigms to survive and offensive ones to fill the stagger meter (so you can deal any worthwhile damage); stats certainly help, but your strategems will play a far bigger role in how battles will go.
Both of these games incorporate stats (which is either the defining or most unnecessary element of the RPG genre), but at the same time find ways to downplay their effect for the benefit of the gameplay as a whole. Even better, they also place a stronger emphasis on strategy instead of just fast reaction-times (though you still can't be too sluggish).
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On the whole , there should be some sort skill involved in playing RPGs. However, it should be moreso oriented towards fast-thinking instead of fast-fingers. And before anyone jumps on this mistake:
That does not automatically mean more complexity!
Really, any idiot can make something more complicated than it has to be (which is why many stat-oriented approaches often scale out of control); more often than not, the right answer is the simple one. The skills an RPG should require is for the player to quickly analyze the situation, and then act (not react) decisively; they're working towards a longer term objective and seeing it through to the end. This offers plenty of ways to throw the players curve-balls and even atypical objectives, creating plenty of depth in the game.
