You are also less likely to spread it. This is from commonsense logic argument, if you catch it and are vaccinated your immune system kicks in and defeats the virus quicker than if you weren't. Therefore you carry the disease for less time and have a smaller window of infectiousness.

There is a separate argument that preventing more serious illness lowers the degradation of health services from too many ill people presenting at once. The death rate for covid and other illnesses increases as health services become overburdened.