United States deaths per million population 262
South Korea deaths per million population 5
This data may be skewed by many other factors, such as nursing home deaths, age demographics, etc.
No doubt it appears in you contact trace someone who has tested positive, and assume those they have come in contact with are infected, then quarantine everyone you believe may be infected it could help. When it comes to the number of cases:
United States cases per million population 4385
South Korea cases per million population 214
The population of United States is 330 million, while the population of South Korea is 52 million. I doubt there is an appetite for contact tracing among Americans to make a major difference. I also doubt any level of government would be effective in developing contact tracing, or using it in a proper manner.
Coronavirus Update (Live): 4,508,704 Cases and 302,324 Deaths from COVID-19 Virus Pandemic - Worldometer
This provides an overview of the amount of contact tracing with is (supposedly) being done by various countries.
Which countries do COVID-19 contact tracing?