Antibody tests or antigen tests? I'm aware of that style of tests for the former but not the latter. You need the latter to do what you describe.
The problem is that their % positive agreement is too low (<90%), and their % negative agreement is based on the PCR benchmark which already churns out a false negative rate of roundabout 30%.
And even their positives are only reasonably reliable (<90%) during peak symptoms. If you're pre, post, or low symptomaticity, its accuracy isnt even characterized (but believed to be low.)
Given all of the above, hyping them up is ... well, ignorant of all of the above, one hopes.
If you don't catch them on the very first day they have enough virus in their body to be above the test threshold, then you'll probably catch them in the next day or two anyway. Also, positive results can be verified with more reliable tests, if you want... or you could just retest with the same test some more, with every consistent result raising one's confidence in the test.
The TWiV episode goes in to detail about how if the goal is reducing the spread of the virus on a population level, then we only really care about a narrow window of infectivity, not about when people are infected but don't yet have enough virus to infect others. The cheap, quick tests they're talking about are good enough for this purpose.
The Abbott Labs test false positive rate was 0.02%, which seems totally acceptable to me
https://www.statnews.com/2020/05/15/fda-says-abbotts-5-minut...
On the other side of the equation, Mina’s point (see paper in my other comment) is that with frequent enough testing, the math of the epidemiology indicates that the sensitivity does not need to be terribly high to have a huge impact on the trajectory of the pandemic
Edit: I see now that you were talking about antibodies, not antigens, sorry
The low %positive agreement generates false negatives, though, not false positives. False positives, for this context, would be fine.