Perhaps not, but I'd expect you'd flip a switch, and then the automated systems would shut it off, or something to that effect.

You expect wrong. Nature isn't rubber padded, and technology isn't friendly magic. You flip switch, control rods go in, the chain reaction stops, the decay heat continues.

So, set it to shut down automatically after 36 hours without operator action, and it will be fine.

It is 72 hours after the full shutdown, that it melts itself down. edit: or to be pedantic, gets outside design parameters; the melt may take another couple days.

Why didn't they have that problem at Three Mile Island?

Because the spent fuel pool didn't boil itself dry. They had a core meltdown, luckily they had full external power and could keep circulating the coolant.

The Fukushima is the TMI without external power: same reactor types, 3 out of 3 melting. The Tokio being evacuated is Fukushima the other time of the year when wind is blowing inland. Ain't a safety feature of the reactor design that it failed in the season when wind is mostly blowing to ocean. At least they did consider evacuation.