OSB is used for one of two reasons in my area:
just builder's grade prepackaged stairs. I can order a set of stairs for a couple hundred and have them installed in a basement in 2 hours by getting OSB (I never do this but could).
in a cheaply built home these could be used as is and carpeted.
I would personally never leave the OSB under the treads unless it would seriously mess something else up. The caveat is if you are using MDF treads, might as well leave the OSB. OSB is just crap and will certainly not take spills and moisture like a solid piece of wood.
Also I would worry that the "give" of the OSB would lead to squeaky stairs. For instance if I left it and used MDF I would just glue down the MDF everywhere on top. If using solid wood I would want to screw them into the risers. I wouldn't want the OSB sandwiched in the middle since I wouldn't trust this to be stable and even.