Your way is exactly what the neocons tried and led to a horrible mess that is still unfolding

- it is not done and gone because people remember things (what do you think fueled Osama, the Taliban, and Al qeuda?) and there are always people who will conflict with us in the world- it isn't possible to chase everyone down. It's been tried too many times in history from the Greeks to the Romans to England to now in some sense America. It doesn't work and the cost is always higher, not lower than peaceful solutions.
And specifically to N. Korea, the cost of going in (especially now) is much higher than not. No one cares if we still have a decades old feud with them- it doesn't make us appear weak because most countries have similar feuds of their own- it is the great power struggle in international politics.