The way to look at it is as a cheap instant filter, something like a bloom filter, that can protect the more expensive tests that take longer.

Your example assumes there's no hierarchy of available tests, and that this test is the only test there is.

What would really happen is those 6 false positives would be referred for a more accurate test. They might miss a day of school but not a week.

At the same time, your more accurate testing pipeline can now speed up thanks to Little's law. There's dramatically less pressure on the system and less backlog, so you have a second order effect that the more expensive slower tests also become cheaper and quicker.

But even if we gloss over all that, and we're only concerned about false positive rate, then this is still much better than no school at all, as in hard lockdown, which has a 100% false positive rate.

Finally, there's the lives saved because of earlier rapid detection and isolation, with corresponding relief for the health care system, leading to increased quality of care and resources available for more severe cases... and so on and so on.

A bloom filter can do wonders for a system, and if this test works it should do the same.