There's a difference between being strict and being wrong. Moreover, if AA was so sure that their interpretation of the original physical document being required is correct, why would they suggest that the OP drive to LAX? Surely they must believe that the same rule applies whether the OP departs from SAN or from LAX. If AA is implicitly admitting that the specific airline has a stricter rule than the country and other airlines do, then AA needs to clearly inform customers well in advance.

It doesn't help with the immediate problem of returning to Switzerland or at least leaving the USA very soon, but the OP should file a complaint with DOT.