I think it's interesting to wonder about the mentality of someone who wouldn't step down after protests like the ones that have been happening. Wouldn't any decent person, whether or not they thought the criticism was reasonable, say "I hear you, it's clear I've lost your confidence, I'm stepping down to allow you to choose someone you trust"?
Barack Obama did not step down after Occupy Wall Street protests, George W Bush did not step down after Iraq war protests, Sarkozy did not step down after austerity protests, Erdogan did not step down after people protested him.
Nobody steps down after protests - welcome to the real world. No need to question their mentality.
South Korea is different. When the war vets and retired special forces protested and completely stomped the anti-riot police.The gov had to obey.
You do not want to piss off south koreans. Ever.
This whole affair has been really insane. I invite everyone to read up on this, it's one of the instances where if this was a movie, if would feel too scripted.