I disagree. You could, but there would be less power behind it. The draw weight of a longbow increases as the string is pulled back. I had a close look at the video, and it looks as if he's drawing the bow halfway at most.
That would be fine against unarmoured troops, probably, or even against most cavalry perhaps (the horse wasn't usually armoured at all) but getting through armour with an arrow fired from a half-drawn bow...
Also, archery tactics depend on bow and situation. It is certainly true that English longbowmen fired in a high arc in most situations, but I much doubt that the arc would be as high in the case of Mongol horse archers. They avoided getting involved in melee by mobility rather than being out of range.
I imagine that bows were fairly often used in ambush situations as well; one or two rounds and then get ready to fight.