I'll start by saying I install tubes/tires on a carpeted surface.
When I get around to that last little bit of tire not on the rim, what I find helps is I will hold it as you were in the video, with the bead being seated away from me, and the unseated section facing up. wrap fingertips around bead area, curling palms around the top of the tire. Push the rim down at an arc away from you and pull the bead up. Even the most stubborn (folding bead) tires snap right on using this method.
I like your way of going around the tire with low pressure checking to make sure the tube is not pinched, and also helping seat the bead at the same time. I will remember that!