PCR is a widely available analytical tool that many hospitals and forensic labs are capable of running. The key seems to be in the choice of which viral genes are targeted and the availability of those needed reagents.
PCR uses a small amount of synthetic DNA, called primer, that's about 10 base pairs long. The primer is designed to only bind to a segment of viral complimentary DNA, usually at the start of the strand for one particular gene. The polymerase enzyme finds this primer-bound segment and begins replication. In this manner, only the DNA you want is amplified, and the much greater amounts of other DNA, such as from the patient, is ignored and eventually drowned out by the replicated viral DNA.
My guess is that the primer is the reagent in short supply that we keep hearing about. Not only that, but in an early CDC test a primer was chosen that targeted a gene non-specific to CoV-2, leading to ambiguous results or even false positives. The South Koreans and others appear to have chosen the correct gene to target.
The South Koreans also have an antibody test that works in minutes, much faster than PCR. These are much like the standard flu test.