When you write "the fuel is extremely hot and takes about a week to cool it to a normal level," perhaps what you mean is not that the fuel is literally physical

When you write "the fuel is extremely hot and takes about a week to cool it to a normal level," perhaps what you mean is not that the fuel is literally physically hot, but that the fuel is so radioactive that it continues releasing a significant amount of energy from spontaneous decay even in the absence of significant neutron flux to trigger further U or Pu fission. When a U or Pu nucleus fissions there are several possible outcomes. Some of those outcomes yield radioisotopes whose half-life is on the order of a week. Thus when you suddenly interrupt a running nuclear reactor, you tend to still have a some energy output queued up to happen spontaneously regardless of any further fission/neutron chain reaction. IIRC, your "week" figure is roughly correct, but not because of ordinary heat capacity, but because of this continuing radioactive decay: the energy output typically continues for days to be several percent of steady state power output preceding the shutdown. (Which is enormously more energy than could be stored as heat in the mass of the reactor at any earthly temperature.)