Earlier quoted context omitted.
Third option: flawed hypothesis. Swabbing the nose or throat and performing subsequent RT-PCR analysis is not an infallable indicator of whether or not someone is free from the virus SARS-CoV-2 or infected with or likely to become infected with the respiratory disease COVID-19. Before 2020 such tests were typically only used as a part of a diagnosis by a medical professional upon consultation, typically also alongsid…
Nose swabs reveal whether you're shedding the particles and thus infectious. It doesn't matter if you're infected if you're not shedding the virus. They are also exceptionally reliable. The home test kit I used had a sub-1% false negative rate.
No, they really don't. Swabbing for RNA picks up gene fragments that may or may not be from infectious virus -- it's why we see positive tests for months after infection in some people.
Swabbing for viral protein is debatably more likely to detect the thing of interest (the virus itself, in some semblance of functioning order), but these tests also have a high false-negative rate (around 10% for the better tests I've seen; I have never heard of a test with a sub-percent FN rate, as you claim). You can be shedding live virus and these tests won't pick it up, either because you're not shedding enough, or because the antibodies in the test don't bind to the protein in your sample for whatever reason.
Either way, you're measuring a proxy for what you really care about. A true test of infection involves taking a sample and incubating in cell culture. Nobody does this, except to validate the original tests and provide clear positive and negative samples. It's slow and orders of magnitude more expensive than even PCR testing. But this is the direct test for infectious virus. Everything else is an approximation.
(Let me be clear, though: I wholeheartedly support the use of antigen tests -- even ones with low sensitivity -- over the insanity we're doing now in the US. It's just bad to misrepresent what they're actually doing.)