Imagine a curve that goes up by two price units in a second, then goes down by one price unit the next second. Repeat that many times. In this situation NumRiseFall will return a streak of one time step for going up as well as going down, suggesting that they be equal, but clearly overall the price is rising on average.

Considering close prices has more random jitter in it, whereas price is the average over the entire bar. Your observation shows that at one bar time-length the price on average is not really random, while below one bar time-length random noise seems to take over.

Time scale of measurements makes the difference here, I think.

And yes, it is true that Close prices are also equally spaced in time compared to price. Still, the difference is that price takes into account a collection of many measurements within a bar, while close is just a single measurement - which is enough to enlarge noise to signal.