CDC really should have a database to log who has been vaccinated and then provide an app that can be downloaded by a person to show proof of vax. Docs would send details directly to the CDC database upon patient vax.
Can docs lie and submit false info? Yep. Can they lose their MD license. Hell yeah. Can databases by hacked? Oh yeah. But the percentage is far less than someone who can easily produce a bootleg fax card from their printer.
This paper card crap is old school and clearly easy to manipulate.