I can at least address this from a physics standpoint, because the video is full of bad, bad, bad physics (not the least of which is using "power" to mean "kinetic energy). Someone else is probably better suited to address the issue of attestation. Mods, if you feel this insufficiently addresses the question, I understand.

First law of thermodynamics - energy lost by performing work is transferred somewhere else. An arrow does not only have kinetic energy, it also has potential energy. Any energy that is lost working against gravity is exactly regained while falling down. The only energy lost is to air resistance. I can't tell you exactly how big a problem this was to medieval archers (arrows are obviously quite aerodynamic but aerodynamics is a very, very complex subject, especially when you factor wind in) but what I can tell you is important - air resistance is, like kinetic energy, proportional to velocity squared. Generally speaking it is for this reason more efficient in terms of conserving energy to have the arrow travel in an arc, because potential energy isn't lost to air resistance. If the arc length is twice as long but the arrow travels at half the speed then you only lose half the energy (since the arrow doesn't travel at constant speed the real problem is a bit more complex, but you get the idea!)

Finally, the gain in range by firing the arrow at 30°-45° is astounding. With no air resistance 45° maximizes range. Apart from obviously being able to aim over the heads of your allies, this allows for a wide range of tactics simply not possible when firing straight ahead. Needless to say, there are situations when firing almost straight ahead is advantageous as well; they are really two different uses of the weapon.