What happened in the dark days of Panem to make it divide into the districts?This might help to find out why they chose to destroy district 13.
The Dark Days were in essence a civil war or a rebellion in which the thirteen Districts of Panem rose up against the oppressive government of the Capitol. The districts were already present before the rise against the government of Panem.
"Then came the Dark Days, the uprising fo the districts against the Capitol." (18).
The Treaty of Treason was the result of the civil war which resulted in twelve of the districts being punished and the thirteenth being anhilated (though later we find out this was propaganda and District 13 does still exist). Part of the punishment was the Hunger Games themselves, which would serve as a reminder to the people of Panem that they were powerless against the might of the Capitol.