Try changing $g$, the value of gravitational acceleration. You will notice that the shape of the path of the rolling ball does not change. The ball does complete the path faster, of course. The qualitative explanation is that if we increase $g$, the ball will roll down faster on the ramp, and so it will acquire a larger horizontal speed. It will fall faster too, but that horizontal speed "compensates" for the faster fall, in the sense that it also goes forward faster. The ratio between horizontal and vertical speed is preserved, and the shape of the path is thus constant.

The explanation below relies on Newton's second law, which is only introduced later in Chapter 9 of The Feynman Lectures on Physics.

Try changing $M$, the mass of the hanging ball. You will notice that it does not fall faster (same with the smaller ball, try changing $m$), only the physical size of it changes. To understand why, we look at the gravitational force $F$, which is proportional to the ball's mass $M$, and the gravitational acceleration $g$. $$F = Mg$$

However, according to Newton's second law, force is mass times acceleration ($F = Ma$). If we substitute this into our equation above, we get: $$Ma = Mg$$ Because the ball's mass $M$ appears on both sides, we can cancel it out completely. This leaves us with: $$a = g$$ Therefore, the acceleration, $a$, which is what actually determines the speed with which the object falls down, does not depend on $M$ at all.