For starters, F = m*a isn't "the equation for force", it's the equation that relates force to mass and acceleration. For the force exerted by an astronaut sitting on the vehicle, the acceleration is the gravitational acceleration, which on the moon is 1/6th of Earth's. His claim that "weight doesn't enter the equation at all" is wrong, F is the weight in this instance.
Gravity would be less relevant in a high speed collision, but they tried to avoid those for obvious reasons. The vehicle's maximum speed was 8 mph, 3.6 m/s, and its loaded mass was 700 kg. We can ballpark the maximum forward acceleration on flat ground using the time required to give it its kinetic energy at top speed...(0.5*700 kg*(3.6 m/s)^2)/(745 watts) = 6 seconds, meaning a forward acceleration of 0.6 m/s^2...about a third of lunar gravity. The dominant force on the vehicle was gravity, which exerted 1/6th the force on the moon...not "basically the same as here on Earth".
As for the motor power issue, the reduced gravity basically meant that climbing a given slope took 1/6th the power, and the kinetic energy lost by rolling up a hill was 1/6th as high. So while the acceleration on flat ground would be the same, the same vehicle would be able to accelerate up a much steeper hill/drive over larger obstacles in lunar gravity, and once moving, the vehicle could roll over hills 6 times as tall using only enough power to counter friction losses.
