Statelessness is often a good thing in designs in many ways, performance, reliability, maintainability, operating cost etc.

As a concrete example for a common use case, websites show your user or first name in the top right corner. It's common to put this in a cookie set on login/logout. This avoids querying the user directory service (from the frontend or a different backend service) on every pageload. Security wise it doesn't matter if the user tampers with it, it's a display only thing. (More often than not it's signed anyway, and the cookie contains other things as well.)

Cookies meant to be presented back to a server can be signed and (optionally) encrypted. This is also a very widely used pattern ("cookie secret" is a good search term for concrete implementations in frameworks).