Check out the two articles linked below. There seems to be a cottage industry producing paperwork for fake negative Covid-19 tests.
Passengers Are Faking Negative Covid Tests In Order To Travel
95% Of Passengers On One Flight Faked Their Negative Covid-19 Tests
I predict that in the future making faked Covid-19 test results and vaccinations will be a profitable enterprise.
Besides the fraud, there are lots of other reasons why requiring testing before boarding a flight would be a really bad idea.