I live in France...
SO I can tell you that Amex CANNOT have access to your Tax return, and Cannot contact your bank to obtain your salary.
They just check if you're on the Bank of France file and the FICP file which excluded you from all type of credit.
Then then will ask you for a copy of your ID cardn utility bill, payslip. And after that amex will send out a letter to your employer to confirm your address, nothing more. I know some people who lied on their application and get approved for the GOld Card.
and for your knowledge, in france they don't have Credit agencies, credit rating and score. none lenders will know that you lie;
if you lied and amex discovered it, at the worst they will closed your account, nothing more