"Private sector?!?" What the...?
Do you really not understand what South Korea did? They got hit by SARS years ago, so the *cough* government prepared well in advance for a similar event. Back when South Korea had a handful of cases, when infected people were identified, they were quickly tested, as were people they had encountered and all their relatives. The *cough* government engaged in a big public campaign, massive testing (free of charge), highly invasive tracking techniques (mostly via cell phones). The *cough* government sends out alerts about anyone who has a positive test, so that anyone who crossed paths with that individual knows to get tested (and voluntarily does so). The *cough* government sanitized the streets in hot spots.
Oh, and the *cough* government encouraged social distancing. They didn't need to make it mandatory, because the public voluntarily isolated itself.
All of this, by the way, only works when there is high trust in the government.... and small hot spots. It can't work if the virus spreads widely.
The private sector had a role... to provide what the government ordered. The private sector was not in any way in charge.
Despite all this, South Korea is hardly free of COVID-19. As soon as they let up, the virus can spread. And if it spreads too widely, then they'll have to do the same kind of lockdowns as every other nation.
So, I really have no idea why you think South Korea had some sort of "private sector" response.