This is incorrect - antigens are parts of the virus - you have them in your body as soon as you are infected. Antigen tests are less sensitive than PCR because PCR's trick is that it multiplies the amount of viral DNA it starts with making it easier to detect. Because antigen tests are less sensitive they *might* miss the virus early or late in the infection when there's less virus present. Your body produces antibodies which attach to antigens so yes, those come later but that is a different test which is relatively rare at this point.
Jamaica and United States are accepting antigen or PCR tests for travel - both test for virus. Antigen tests are faster and cheaper than PCR but less sensitive. Neither Jamaica nor United States accepts antibody tests but nobody will offer those to you when you ask to be tested for covid.