I would go with the veneer'd flooring material. If you resaw thicker stock, I would suspect you may have a problem with the pieces splitting along the grain if you make it too thin. How would you attach the pieces? I would certainly want them SOLID for stairs. That being said, I am thinking glueing them down.
Good quality engineered flooring should have a thick enough veneer and tough enough finish that wear should not be a big issue.