Replacing oak stair tread

Got a picture? It's hard to see it from here.

That sure looks like the treads run under the skirts which would make removing the tread a real challange.
If was mine I would first try cleaning out that gap, use Tite Bond II glue and counter bore hole for screws and cover the screw heads with white oak plugs, sand and reseal.
By using 3" ceramic coated deck screws it should never move again.

Hard to tell what were seeing taken that close up.
Just to make sure were talking about the same thing, the skirts are on the sides and the riser is at the back of the treads.
If there not not under the skirt you should be able to just use a flat bar or a rubber mallet to lift up the front of the tread, may be easyer once it's up to cut of any fastners with a sawsall so it can be slid out.
Use the old one for a template for width and depth.
Just use some constrution adhesive on the stringers and long galvinized finished nails to reattach it in predrilled pilot holes. I use galvinized because of the rough finish that holds them in better.

Most oak treads are Red Oak and that is what your's look like. White Oak has a brownish cast and has a closed grain. Red Oak has a slight pinkish color and has an open grain or little pin holes in the end grain.