The idea that the plane was unable to leave ground effect is not quite right, in fact it's the opposite.
Ground effect results in the plane having slightly more lift (and less drag) at very low altitudes close to the ground, and it typically comes into play when your altitude is less than the wingspan of the aircraft, this is why it is sometimes described as a cushion feeling in the last moments as a plane descends down to ground level.
So for example, let's assume you are stable and approaching the runway at a constant rate of descent, lift and drag will be very much constant. Suddenly however, as you pass down to an altitude less than your wingspan, the aircraft gets a slight increase in lift, and a corresponding decrease in drag (specifically, induced drag).
As a result, your rate of descent will decrease due to the increase in lift, and your airspeed may also increase a little bit, which can give the sensation of the plane floating just as it is about to land. I personally found this a good source for understanding ground effect and induced drag:
Going back to the video then, the airplane is actually generating slightly more lift during the take-off roll thanks to ground effect, which essentially is helping to push the plane upwards (hence why I said it's quite the opposite above). In fact, the pilot states that he managed to gain a small amount of altitude before feeling that he had suddenly lost lift and deciding to abort, and that is likely due to primarily the weather conditions, but possibly also supplemented by the fact that he did in fact leave ground effect, and no longer had the benefit of the increased lift (and reduced drag), and hence did not have sufficient lift to sustain the initial climb, especially given the conditions.
However, I think you'd have to say that the main cause of the aborted take-off was the extreme wind conditions which did not allow them to generate sufficient airspeed required for the climb. As the pilot states, the wind changed dramatically during the take-off roll, and you can see the airspeed indicator moving around a lot too.
So in summary, changes in induced drag (which is essentially what causes ground effect) are a function of altitude vs wingspan, and is not directly related to speed.