Why Do Nuclear Power Plants have to meltdown when a catastrophe strikes?

I don't get nuclear power plant meltdowns. I have heard the explanation that shutting down a nuclear power plant is like turning off a hot stove. Unlike a conventional stove, If the nuclear power plant does not cool down properly stuff happens such as rods heat up rather than cool down and radioactivity can be released into the air after a meltdown if the five foot thick cement enclosure explodes... but something still doesn't make sense to me.

I've seen dvd players where even if the tray is extended out, when one turns off the power, the tray retracts before the power actually goes off. Is there no way to create a nuclear power plant where the rods used to create nuclear energy automatically retract away from each other in an emergency situation? Would this not prevent a nuclear meltdown from occurring whenever a nuclear power plant has to go offline in an emergency?

Although even if this idea makes sense, and it was incorporated in the future to prevent nuclear meltdowns, is the planet really better off with more and more nuclear power plants?

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