The rapid antigen tests have a significant problem with false-negatives: the BMJ reported that Innova tests have 49% sensitivity.

That makes it pointless to use as a border control - for every two cases, at least one gets through. (Or, potentially much worse, under this system: I throw away my positive tests and show whatever false negative one I have).

You ideally want to be conservative and permit some false positives at an effective border control, because you don't want to risk infectious cases getting through and causing an untraceable national breakout.