It depends on the error rate of the test. If it's close to 50/50 then obviously the test is worthless. But these tests are much more reliable than that, so a population scale the wrong results will be corrected with more testing.

I don't have the analysis in front of me, but I encourage you to listen to that TWiV episode yourself and listen to the panel of virologists discussing this. They were initially skeptical also, but after questioning the plan's author they were eventually convinced and became excited because the plan could actually work.