Good question. The way Mina has described it, the virus is growing exponentially at first - so quickly that a person could be below the threshold of detection (say a few hundred copies of the virus) in the morning and have millions of virus by the evening. So while the rapid antigen test would miss that person the first morning, it would catch them the next morning. At that point, the person is still asymptomatic, but they are infectious.
So maybe the rapid test doesn't catch them on day 1, but it catches them day 2. If they got a PCR test it could catch them on day 1 because it's sensitive at those lower virus levels. BUT the reality is, we are still woefully under-testing, and the person wouldn't know to get tested until they were symptomatic anyway. Likely, they won't express symptoms until day 3 of the infection, get an appointment for a test the next day (day 4), and get the results back in 3-5 days if they're lucky (day 7-9 of the infection). In other words, that person might not have a positive test and so might not do anything about their infection for 9 days because it took so long for them to get access to a test and then get results back.
With the rapid test, no, it wouldn't catch them on day 1, but it would on day 2, immediately prompting isolation and stopping the train of transmission. That is why, in our current testing environment, a fast but less sensitive test is superior to a slow but more sensitive one.