That's an interesting video, but it also illustrates my point.
Folding identical sized towels made of the same material takes that robot a long time. It would have no chance to iron anything. Note how it smoothed out the creases in the towels, and then just think how many variations it could need to do something similar to shirts, bearing in the mind the shirt could be made of wool, cotton or silk etc. The robot would need many more sensors to detect material types.
I imagine it would need more 'intelligence just to check pockets were empty or not, than it takes to fold a towel.
Either way, the video is still impressive