OK. I hear what everyone has to say so far. And, I agree. How about when it comes to agility? I mean the ability to quickly act on or react in tight situations. In particular, I'm curious about whether everyone here thinks an ordinary strong gymnast has enough to go from a mostly pike/tuck position into an upright and back on demand (multiple times) while the body is moving up to 40/50 mph (or higher). My understanding is lots of superman helps to condition for this situation. Of course, there is no such thing as too much core strength for a everyday gymnast.
Being able to recruit muscle power (well this is my understanding) has more to do with muscle composition. Being strong in those positions is important, because there is a connection between the neurons firing and you need to maintain the most efficient shapes in order to use the equipment well and accelerate or maintain power in the skill.
Most high level gymnasts are better sprinters and jumpers than endurance runners, because they have a lot of fast twitch muscle, relatively. This enables the muscles to "fire" in fast, short bursts that do a lot of work, but they tire out more quickly.
I'm not sure what skill you mean...a double back? This is not something most kids who start competitive gymnastics or even finish compulsories will get, so I guess in that sense "ordinary strong" isn't enough. But if a kid is strong and able enough to get a lot of the beginning optional skills, I tend to think most of them, if they want it, could build the strength component for it. Now other issues come into play obviously, the mental component or bad habits that contribute to poor technique (so this is where being strong in the correct positions and body aware to recognize and apply them would come in again, I guess).
We need to use a lot of exercises to condition for these movements...stretching and snapping. When twisting is added, need to be able to maintain tight shape there too, side core conditioning. Turning on bars. I'll be the first to admit that my knowledge of conditioning isn't "formal" by any means, but I try to more or less figure it out from those who know more, because there's an efficiency thing to this. Shawn Johnson's coaches apparently have great knowledge of efficient body prep. Obviously we're not all training for the olympics and don't need to be super efficient, but it's still pretty important and can make the difference between the same training hours and better gymnastics.