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