I live right next to Slatington PA.
Which is a few miles from Slatedale which still has an active slate mine.
The "Slate Belt" region around Bangor is only a few miles away and also has an active slate mine.
I may be prejudiced when I say that slate is very versatile stone for many uses.
It can be "machined" in many different finishes.
It is durable to almost everything except for hard, heavy impacts.
All around my town are houses with slate roofs that are closing in on 100 years old.
Granted, some are better than others with maintenance throughout the years and some that have been let go and are being sheathed and covered with shingles year by year.

There are even slate curbs along some roads that are still in perfect looking shape.
I will say that anything close to smooth slate with a finish on it will be slick. Textured slate would be better for this.
While slate is pretty dense and good at not getting deeply stained, I would still seal it somehow in a bathroom application like this.
Often times inside (like a bathroom) can be a worse environment than outside with soaps and everything else that happens in a bathroom.