The coiling aspect, for me, comes from Sanchin and is meant to work with Posture for Power Transfer (Atifa) and proper mechanical application, ie Tension, at the very end for just a split second (Chinkuchi) drawn from the ground up through the hips and the core first (Gamaku).
Having gone through this slow a few times just to feel the timing and the nature of each within these two very different Kata it is like a bow-string pulled taut then released and snapping back (taut again for just a split second on the opposite end).
The stances in Karate are too emphasized and should really be de-emphasized in favor of looking more at the actual movements themselves rather than some cool looking position that really has no meaning at the 'end' other than where you should end up.
Gekesai Kata was meant to be a training Kata, very basic, but also very brutal if understood correctly, the name gives you exactly what principles and characteristics the Kata focuses on... To Attack and Smash or Destroy.
As Such the Kata is full of various attacks that are straight forward Punches with tension applied for proper effect, with smashing attacks like Elbow Strikes (which also double as entering head guards), upward smashing attacks (Jodan Uke) and downward dislocations/breaks (Gedan Barai), Mid-level manipulations (Chudan Uke) in set up for a nicely placed Heel-kick to the knee (or there-abouts) of the lead (or rear) leg of the opponent BEFORE delivering the Elbow.
At the beginning is that coil/uncoil, the delivery is very different depending on the blow. Punches are meant to act upon the water aspect of the body, causing Hydrostatic Shock and recoil in the opponent, thus the slight and quick tension on the end, Elbow Strikes are meant to Smash... I am sure you get the idea.
Seipai is a very different Kata in its' characteristics, its' Nature is not quite so direct and aggressive, although no less effective.
It plays more on the coil/uncoil element throughout with A LOT of circular movements and linear movements that play off the circular (similar to Gekesai in that respect, but quite different at the same time).
The beginning coils up and drops down with a caption and strike/lock on the arm/neck area following through with a Shihonage-like/Aikido-like techniqe that leads into a break (when I say Shihonage-like I mean to say it can be applied as such, but can also be applied to the head and would make a lot more sense considering it ends with a drop into Shiko-position).
Lots of coil/uncoil in that movement, then it leads into an interesting sweep-attack with the foot into a weird back-leaning position, one arm up, the other sweeping down, both open-handed.
It can be a redundancy follow-through from the previous movement as almost an Ashi-Barai where the foot does not leave the ground, one hand pulls back as an anchor while the other sweeps through to take down.
In all this, the main point I always come back to is the Hara.... My Sensei used to say to pretend I had no arms and that all movements are connected directy to the Hara.
These things are far less effective without a thorough understand of how the Hara is applied in each, how it moves, raises, sinks, spins around the axis, ect.
I once watched a video of a Karteka against an Aikidoka and they used some sort of computer program to trace the movements of their center through Shihonage.
The Karateka tended to rely more on his strength and his Hara stayed on a straight line, almost unused, while the Aikidoka did not have brute strength to rely on and utilized a linear movement combined with a well-timed drop of his Hara to apply the technique.
There was a HUGE difference in the movement and it says a lot about the state of Karate today... Many need to go back to the floor and REALLY look at what they are doing with their Hara, take some lessons from Aikido and Judo.
In the end there are really only a couple of things to master... A few techniques and how to move from your Center...
You do not need style to do this, you do not really need anything but a few movements and you can go from there.
Seipai and Gekesai are just fancy ways to go about it... Really my goal is to simplify and shorten so that the focus is only on a couple of things.