Other theories are available
My favourite theory references that not only do Irish dancers keep their hands by their sides, but they also aim for a very straight and non-mobile upper body. Apparently this stems from more puritanical times when dancing was frowned upon if not prohibited.
When dancing only, the legs/feet move. Which means that you can dance in the pub and the authority figure wandering by and looking in the window only sees the upper bodies, which aren't moving