It's in South Korea where there is an abundance of tests. Where tests or people to administrate them are scarce, we have to prioritize those with symptoms.
There is an argument to me made for the exact opposite. Those who are already sick will be treated according to their symptoms, a diagnosis has little bearing on the treatment they will receive and therefore low value. Meanwhile, proper random sampling of the population will give more accurate prevalence numbers and trends, allowing for much better epidemic response.
South Korea (and Singapore and HK etc.) had a specific policy of contact tracing instead of random sampling, which works if you start it early enough and can actually keep ahead of suspected contact spread.
"Those who are already sick will be treated according to their symptoms, a diagnosis has little bearing on the treatment they will receive and therefore low value."
It is not low value for the healthcare providers, though, who can spare measures against getting infected to those who actually have COVID and thus both save time and scarce resources.
The cost of false negatives would be too high. Hospitals have to treat all respiratory cases as potential COVID right now, even if initial tests are negative.
Sick people should be tested when they require medical intervention as it guides their care. It would be a shame to assume COVID when it might be something treatable.
I'm sure the decision tree is complex, and there are branches where what you say is absolutely true. However, given the testing kit bottleneck, if identification of "something treatable" is crucial, it might be better to test for those treatable diseases that are likely given presenting symptoms.
In the extreme case, if you require a ventilator right now or you'll die, nobody needs to wait on verification of the specific viral cause to begin treatment.
