Cider pressing ?
Can't see it needing all that slate though.
A really stable base for the press and some channelling for the raw must to be put into barrels, and I think that's about it really.
Good worked slate, in really sizeable slices....not usually used for flooring or architectural apart from roofs, shelves and something you really need to be level.
We know that slate was used for pantries, for dairies, basically chill areas that needed clean surfaces. They didn't have plastic workstops in the past. They did have marble but it's porous and easily stained. Wood is good, takes scrubbing, but slate is preferred.
Middle of this page, fifth photo down.
www.beechvillage.org.uk
Victorian Welsh Slate Shelves which were used as shelving from the Public Records Office in Chancery Lane, London.
suffolkreclamation.co.uk
Audley End, country house, it's larder....
Does anything look familiar about your pieces compared to these ones ?