We do have problems with that approach, though.I think without a vaccine or effective treatment, a simple home test with a quick result is the answer. A test which people can easily get a supply of and self test 2 - 3 times per week. I suppose the scale of production necessary would make it affordable for people to do this.
If people want to appear ill, what is to stop them lying?
If people want to appear NOT ill, what is to stop them lying?
There will be all sorts of issues where there are perceived advantages to individuals to lie either way. Unfortunately, people will lie unless there is some sort of proof of having taken the test and the result being as claimed. I can see much effort going into developing techniques to stop such lies but they'll probably all have blooming great holes.
Otherwise, yes, excellent approach.