But isn't it the actuall ke that helps makes the arrow more stable.
Yep. Here I is.
No. Momentum is actually what makes an arrow more stable. By definition, momentum is force applied in a specific direction. KE is a non-directional factor. The more momentum, the more it resists any change in it's direction. That is why a heavier arrow is more stable.
Increased FOC is another factor that enhances stability. Aircraft that are too light in the front end will not fly at all. Drag on the back end of the shaft enhances stability. Also, spin around the longitudinal axis of the arrow is a huge stabilizing factor.
Bullets have plenty of KE and most of their weight is in the back (back of center balance). But if they don't have enough spin to stabilize them, they begin to tumble.