What is noticeable about South Korea was that they were a high profile example not of social distancing, or hygiene but of testing, and tracing.
The UK health authorities know full well that testing and tracing can be effective and furthermore they have the capacity to do it. I think they have chosen not to because they see some benefit in insuring against a 2nd wave by maintaining the infection rate. This maybe a nonsense strategy. At the moment I think it's a conceptual argument. No one wants to lockdown their country and see people die but there are two strategies to fight the virus: total suppression and controlled infection. Mass testing separates the two. Antibody tests may reveal whether herd immunity is attainable and at what cost. Naturally we hope that as many New Yorkers as possible have antibodies to the virus but I suspect they will be found in only small numbers of people.