if the star had exploded in a supernova after exhausting all it's nuclear fuel, then there would have been nothing left to form the pulsar with

supernovas are caused by the starting of another stage of fusion, where the star is large enough for it to be a supernova. every star goes through stages of fusion, starting with hydrogen and moving up the periodic table. The Nova when they have reached the end of the common elements to fuse together, not from a lack of fuel, but because the fuel can't be fused any more in a stable reaction.