There is little purpose in using tests to check asymptomatic children to see if it is safe for them to come to school. When children are infected, most are asymptomatic, and the mortality risk is lower than for the flu. While adult-to-adult and adult-to-child transmission is common, child-to-adult transmission isn’t. Children thus pose minimal risk to their teachers. If a child has a cough, a runny nose or other respiratory symptoms, he should stay home. You don’t need a test for that.
Sweden was the only major Western country that kept schools open for kids 15 and younger throughout the pandemic, with no masks or mass testing. How did it turn out? Zero Covid-19 deaths among 1.8 million children attending day care or school. Teachers didn’t have an excess infection risk compared with the average of other professions.