It's not so simple, as you think.
North Korea is brutal dictatorship since 1945 when Kim's grandfather become first president.
In the past they already invited South Korea, but America saved South at that time, by sending military troops.
Only reason why North Korea never invited South again is because of American military support last 60-70 years.
You can't blame the North Koreans for everything. I have noticed that in most of the cases, they are reacting to the provocations created by the Americans and the North Koreans. Perhaps the situation will improve, if the South is allowed to directly deal with the North.