What public services does the government forbid you to access without a Covid vaccine? I am not aware of any.

If you are talking about private services, then there are many examples where papers are needed:

• buying alcohol (ID)

• entering a nightclub (ID)

• taking a plane (ID)

• opening a bank account (SSN/ITIN)

• renting a car (driver's license)

• buying a gun (proof of residence)