I don't think it's true to say no new heat is being added. The uranium fission process has stopped, but there are secondary radioactive elements created by the decay of uranium. These elements will continue to decay, as such the core is still producing the heat. Though eventually this heat will decline to a safe level, provided there isn't a meltdown beforehand.

The design oversight seems to be that they depend on external AC power or diesel generators for backup. Surely there should be an emergency generator that can run off the heat that the core still produces in order to at least partially operate the cooling system. Much like aeroplanes have a small dynamo that runs off a tiny wind emergency wind turbine so they can still operate the hydraulics with no engines running. A saving grace in the case of Gimli Glider (http://en.wikipedia.org/wiki/Gimli_Glider).