Staining and grout removal aside, are there other disadvantages to using natural stone tile for bathtubs?
err. . .it's not completely waterproof!
Using natural stone in the bath tub is not that advisable. removal of grout and stain is not a problem if the granite is well polished you know. I want to tell about one issue only which I have seen many places. You know if you compare with ceramic tiles, the natural stone is a porous material, I mean there are many invisible holes on it. The effect is that stones will absorb water and it will spread into the foor and it wil make problem if your bath is in the first floor or above. See the problem if you are on a floor above the ground floor, you will see water dripping from the bottom area of the tub. In this case, if you are particular in going with natural stone, you can make it somewhere out side the building with separate foundation.
For the outside sounds really neat... and if inside we are talking well drained sand floors with slope to outside(remember New Orleans... it was the levees designed to keep city dry that flooded it!) Expense would be another good reason besides you do know how hard ring around the tub is on vinyl/fibercase/porcelin tubs NOW imagine porous rock... But as cost no object allow larger size(for rock sides/bottom) build water tight container(well supported- no joke intended) with drain/ overflow and then place rocks and grout(optional) away... you are doing this ground level or then ALSO consider weight and add new truss supports etc. Will this new tub be inspected by city/county? They may have something to say... or if you have plans to ever sell... Looking better and better outside? Concrete pond ! Yeah I know,,,