but i think what you're asking is why do we have fairly compact flywheels rather than longer, more extended "tails"? and the answer is probably that it makes more engineering sense. you can increase the angular momentum stored in something by making it "bigger" or by spinning it faster. i imagine it's a lot easier to spin a small flywheel to thousands of rpm than to make a "tail" (spinning at a very low speed) thousands of times larger.
also - unless i missed it - this video doesn't really explain all of it. by itself a cat's tail isn't nearly large enough or rotating fast enough to spin a cat's body. the cat is also very "intelligently" (presumably "designed" through natural selection) doing much of the rotation while in a U shape. that reduces the net angular momentum considerably (the two "arms" of the U effectively cancel).
and, from what i remember elsewhere, they also reduce the amount of rotation needed by rotating to only get one pair of legs perfectly aligned. as long as those touch down first there is a little time (i admit i am not sure it is sufficient) to then rotate the other legs while "holding on" to the ground.