At least one company in South Korea, Seegene, has produced many tests and it is this aggressive private sector production that is largely reasonable for South Korea's effective testing. There is also a large demand for Seegene's tests around the world indicating that WHO tests are definitely not in sufficient supply.
The private sector stepping in is the best way to get enough tests out to everyone who needs them.
This is March 1st press conference from South Korea: https://news.v.daum.net/v/20200301162614985. It says Seegene accounts for 37.8% of test volume. The context is that Seegene (and only Seegene, other companies have other supply chains) had supply chain problem. As I understand problem is now resolved.
The private sector stepping in is the best way to get enough tests out to everyone who needs them.
https://www.cnn.com/2020/03/12/asia/coronavirus-south-korea-...