Polyshades is going to be a bear to get applied evenly on a floor I would think. It dries quickly, so it is easy to get overlaps that show up as dark streaks. A quick search on this or other fora will show it to be widely hated.

Howard is correct. If you want to do this, do it right or you will be just making it harder to fix down the road. It needs to be evenly sanded--otherwise defects or variations--such as the places where you scraped off ceiling paint--will show up like a sore thumb. By the way, the old stain is essentially a finish--it contains a very thin varnish that is needed to bind the pigment to the wood. It is this old material that can interfere with the ability of new finish, especially if it is has polyurethane included , to adhere properly.