If you look at the actual testing data for antigen tests [0], the LoD for them is very high compared with PCR. This means a person needs a high viral load for tests to be effective. For example, the BinaxNOW kits require 140.6 TCID50/mL [1] vs 0.012 TCID50/mL for a PCR test (I picked a random one here) [2]. The FlowFlex tests are even worse, requiring 2500 TCID50/mL [3]. I personally purchased a few of the Lucira tests (not affiliated I promise) because they're more similar to an at-home PCR (they use LAMP) and so the LoD is lower (0.9 TCID50/mL) [4].
[0] https://www.fda.gov/medical-devices/coronavirus-disease-2019...
[1] (pdf) https://www.fda.gov/media/141570/download
[2] (pdf) https://www.fda.gov/media/142193/download