Everything you say is true, except that mostly doctors are not busy collecting data, but rather treating patients.
You may have intended to link a different paper; the one you linked is about Gansu in China, not Korea, and does not mention rt-PCR, false positives, viral RNA, or reinfection or recurrence. It does mention "secondary infection" and "secondary cases", but, as it explains, that means people who were infected within the study region by other people ("indigenous"), rather than coming into the region from outside ("imported").