Pretty simple to answer this. Passports work. Car safety regulations work. Seatbelts protect you.
COVID vaccines, however, are not effective at preventing the transmission of COVID, this is well established science at this point. So, that said, why the mandate?
The vaccine mandates prevent unvaccinated people from accessing areas of higher risk transmission. Unvaccinated people (particularly the elderly) are more likely to end up dead or with a serious hospital visit if they catch the disease.
At the moment hospitals are overwhelmed and so at this point the vaccine mandates are about keeping hospital admissions from exploding.
COVID vaccines, however, are not effective at preventing the transmission of COVID, this is well established science at this point. So, that said, why the mandate?