It has almost a 20% of false positives. If you have a class of 30 students, approximately 6 of them will get a false positive on Monday and loose the rest of the week just in case. By Friday, you will have only 10 students...
The next week you will have the same problem ...
The next week, people will start to ignore the test.
And this assuming the students only take one test per day. If they also get tested in the bus and the cafeteria and the supermarket, ... the number of people without a false positive will much lower.
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.
Are the false positives IID, or are they correlated to each other in the same individual, i.e. a false positive will likely test false positive again, a true negative will likely never test false positive? This is the kind of critically important stuff in stochastic processing theory that medical-field statistics never seem to care to report on.
If it is the former (IID), and let's say P_D = 1.0, P_FA = 0.2, it is an extremely easy problem to solve: Just have each student take 3 tests each day, which will reduce the overall P_FA from 0.2 to 0.008. Or 4 tests for 0.0016.
If it is the latter, you will only lose 6 students for the whole week; you'll have 24 students left on Friday, not 10.
Do you think the false positives are per-recording? In that case, you could just do a few more test coughs with the same person to double check, and get that false negative rate down.
If false positives are per-person, then your scenario won't happen. It'll be the same 6 kids for whom the test never works right.
So with all this in mind, you'll have to come up with appropriate norms around the results. You could call it "okay" vs "suspect" instead of negative and positive. Maybe there's a lowered-risk version of activities for people who are "suspect" that day. Maybe they don't go to the gym that day, maybe they sit in the isolated booth in the classroom, whatever. But then, they need to take a standard test that night to return to school the next day. Or as someone else mentioned, a rapid-test at the nurse's office.
The next week you will have the same problem ...
The next week, people will start to ignore the test.
And this assuming the students only take one test per day. If they also get tested in the bus and the cafeteria and the supermarket, ... the number of people without a false positive will much lower.