Is blood from a fresh corpse viable for use in transfusions?

If so, and blood is routinely drained from a body in preparation for burial, why not drain it earlier, at the hospital immediately after death? It seems like that could make a pretty big dent in the available blood supply. Would it work? What are some possible obstacles in this proposal?

Also, if one is an organ donor, do the doctors ‘harvest’ and use the blood as well as the organs?