Pilots are already treated differently by government mandates during the pandemic. Pilots aren’t allowed to fly within 48 hours of taking a vaccine shot due to potential side effects. And pilots are not required to wear face masks while in the cockpit.
The Biden administration executive order directing OSHA to introduce a vaccine mandate on companies with 100+ employees includes an expected alternative means of compliance in the form of regular Covid-19 testing. While United Airlines is not permitting this, a fear of losing pilots would seem to be enough reason why American Airlines would embrace this alternative.