international travel – Why do countries check if arriving persons are vaccinated and not if they have antibodies?

Commercially available antibody tests are not intended to be used to determine vaccination status, aren’t tested to be effective for that purpose, and are known to produce negative results in vaccinated individuals. The test detects specific types of antibodies depending on its design, but that’s not the same thing as a general measure of immunity to COVID-19. As the US CDC explains:

Antibody testing is not currently recommended to assess for immunity to COVID-19 following COVID-19 vaccination or to assess the need for vaccination in an unvaccinated person. Since vaccines induce antibodies to specific viral protein targets, post-vaccination serologic test results will be negative in persons without history of previous natural infection if the test used does not detect antibodies induced by the vaccine.

In other words, the antibodies your body makes in response to a vaccine may not be the same antibodies the test is looking for.

Or see Can antibody tests tell you if a COVID-19 vaccine worked?:

A vaccinated person is very likely to get a negative result from a serology test, even if the vaccine was successful and protective. That’s because different serology tests detect antibodies to different parts of the virus.

Some tests detect antibodies to the spike protein of the virus, which are produced in response to viral infection or the vaccine. Others detect antibodies to a different part of the virus called the nucleocapsid protein, which are produced in response to infection, but not by the current vaccines.

MD Anderson’s Blood Bank uses an antibody test designed to detect antibodies to the nucleocapsid protein, which means donors who have received the COVID-19 vaccine will likely receive a negative antibody test result.

As a result, a requirement for an antibody test would not serve the intended purpose, as some vaccinated travelers would have negative test results despite being well-protected, as the tests aren’t designed to detect immunity from vaccination (though this depends on the exact combination of vaccine and serology test used).

Some countries do offer exemptions from their testing and/or quarantine requirements for travelers with a recent (usually within a few months) documented history of COVID infection and recovery, which does cover the case you mentioned where someone has a degree of natural immunity from a previous infection.