It’s generally hard to have a “false positive” in a PCR based test unless the sample was contaminated with the material trying to be detected. False negatives would be much more common.
Also, even a few percent chance of error in forensic DNA matching is a huge problem. If a virus test is wrong a few percent of the time it's still useful.
[Actually, the article contains the answer to my objection: they tested for this. The test was developed by explicitly looking for primers that give positive results for known-positive COVID-19 patients but give negative results for other test subjects with common colds or other respiratory illnesses. So assuming they had enough training data, this should already be factored in, and the test should remain accurate. Posting this anyway in case other people are thinking along the same lines...]