I'm not an expert, but a couple of my thoughts for what they are worth....
I can see that a wet clay side slope will not be the easiest surface to drive on - let alone any running water issues! That said, around here gravel on a slope has to be chosen carefully as it will tend to move down hill and all end up at the bottom of the slope.
Where does the water come from and where will it go?
The risk of the pipe running the length of the drive is that it will get overloaded and then cause erosion at the point of exit. I often see erosion/ drainage gullies cut at an angle across the track to divert smaller amounts of water to one side. A simple version is just a moderate sized log. This video shows someone has used road barrier sections across a forest track, which seems to have been effective.