[di]

Nope. The private key is generated within the client each time a signing event occurs, and that's what is used to sign the artifact. It doesn't come from the certificate.

The certificate just binds the public key to the identity at a given point in time, in a public way. This certificate is generated every time you sign something, and is put in the transparency log.

There's a walkthrough of the process here that might be helpful: https://www.youtube.com/watch?v=jdf-gNYg0fw&t=494s