Sounds like you don't understand the concept in the video.
It's not a flexible case. It's a single flexible screen inside of a hinged rigid case. There only needs to be a single rigid battery to power it.
Here, I made a side view derived from some of Samsung's patents:
View attachment 634785
Yes, it'll have to be a bit thick to accommodate the room they want to give the fold radius, to prevent a destructive sharp crease. Many of their patents deal with clever hinge designs that maximize the radius while being as unobtrusive as possible.
So, at this point in the technology, I don't think it's intended for teenagers who put phones in slim jean pockets.
But for those of us with bigger pants or shirt pockets, or even business coats, it'll be fine at this point in time. In the future, no doubt it'll get thinner and able to have a sharper fold.