But then every rotation would break production, wouldn't it ?

rotations are usually two phased. Add new secret/credential to endpoint, and both new and old are active and valid. Release new secret/credential to clients of that endpoint, and wait until you dont see any requests using the old credential.

Then you remove the old credential from the endpoint.

Ideally, you can have a couple of working versions at any given time. For instance, an AWS IAM role can have 0 to 2 access keys configured at once. To rotate them, you deactivate all but one key, create a new key, and make that new key the new production value. Once everything's using that key, you can deactivate the old one.