The millions of illegals were not tested for any diseases before flooding across the borders for many years now.
In the Ellis Island days they even watched for some sexually transmitted diseases by doctors visually observing and doing some things wearing rubber gloves as described in World of Our Fathers: The Journey of the East European Jews to America and the Life They Found and Made By Irving Howe (2001).
They should be testing migrants for much more lethal diseases than covid, such as drug resistent TB, typhoid, plague, Syphillis, HIV, etc.