To add to what domo and 4hp have said the protection against severe symptoms is the part that is nearly guaranteed (link with statistics of different vaccines, 100% against severe disease defined by the CDC and 95% defined by FDA. Even the new study that brought the efficacy down from 95% to 84% after 6 months still had the efficacy against severe disease at 97%. It is by no means 'only' for protecting ones self, hence the reason governments all over the world are pushing for people to get vaccinated to stop widespread cases, but the reduction in hospitalization obviously is a huge bonus as well as we all know hospitals all over the world were completely full pre-vaccine.
I'm not sure if the vaccine cards are the right way to do it as, at least in the us, it disproportionately affects black communities which are understandably skeptical of vaccines after the horrendous 'human experimentation' with syphilis. However, if it leads to more people getting vaccinated and stop the second wave with delta variant I can definitely see the reason they are doing it. NYC for example was fucking ravaged by covid so it's totally fair they're being extra cautious now imo