Yeah fakes are probably already circulating. Plus I don’t know anyone who actually carries their card.
Also, what if you lost your card? If you got vaccinated at one of the many pop-up clinics, there’s probably some work involved in getting another card.
I don’t know how accurate the recordkeeping has been and I don’t know how good of a national database there is. My wife’s doctor had no idea she had been vaccinated so clearly no info was sent to our PCP.