I've saw several at shows but never saw one in use. From what I've read they has their pluses and minuses. Good on slopes and could be lowered with a rope easier then an electric wheeled mower. I think many had 2-cycle engines so you did not need to worry about oil starvation from being tipped too far to the side.
Watched a guy using one of those Flymos on his steeply terraced front yard once. He was swinging it back and forth, just like in your video, but at the end of two ropes, dropping it a little lower with each pass. Looked strange, but seemed to be getting the job done.
It seems to me that it would take a lot of downward air movement to make the mower hover in the air like that. And with such a current of air blowing downward it seems that the grass would lie over, which would make it very difficult to achieve a good cut quality.
I guess I should get one and test it for myself...
Yeah, the down draft to float it would seem counter productive considering many mower designs tout their lifting and vacuuming ability. Likely would work best on short stiff bladed varieties of grass but I think their real nitch is mowing where you could only safely go with a string trimmer otherwise.
I had a 2 stroke Flymo for my back bank. Too steep to walk, so I did the rope trick. They work well, though I thought you could only find them across the pond now. There were electric and 4 stroke models in the day. Limited width on the cut, and scary to see a plastic deck.