Is it unlikely that Joseph would have buried Jesus since Jesus' body would have made him unclean?

Joseph likely had assistants who helped him, as the phrase "where they laid him" in Mk 16:6 suggests. That Mk 15:46 says "he [Joseph] took him down" does not necessarily mean that Joseph literally got up there himself and pulled out the nails. As has often been pointed out, the society of Jesus' time considered an action that a person had done through emissaries to be his/her own. Hence, Joseph did not necessarily defile himself by performing the burial.